Question 1Coordinate geometry
The point P lies on the ray that starts at the origin and passes through (4, 3). The distance from P to (8, 1) is 5. What is the greater possible distance from P to the origin?
TMUA Coordinate Geometry — Practice Questions by Topic
These are SummitPapers original questions, not official past paper questions. Official TMUA questions sorted by topic are on TMUA past papers by topic.
- MM3Coordinate geometry
- 98 questions59 on Paper 1 · 39 on Paper 2
- 9 free solutionsThe rest show the correct letter only
Covers: circles, tangents and chords, lines, midpoints and perpendicular distance, parabolas and the distance from a point to a curve, reflections and loci, triangles in the coordinate plane.
What this topic tests
Coordinate geometry here means lines, midpoints and perpendicular distance, circles with tangents and chords, and parabolas, including the distance from a point to a curve.
How it is assessed
This is Section 1, on both papers in this set. Each question has five options. A paper is 20 questions in 75 minutes, with no calculator.
This note is written for SummitPapers. The official list of what can be examined is the specification, together with the Notes on Mathematics and, for Paper 2, the Notes on Logic and Proof.
Key methods
A line
Two points fix the gradient, and the gradient of a perpendicular is the negative reciprocal, when the gradient is not zero. The midpoint averages the coordinates.
A circle
A tangent is perpendicular to the radius at the point of contact. A chord’s perpendicular from the centre bisects the chord. Expanding (x − a)² + (y − b)² = r² and completing the square are the same circle.
Common mistakes
Using the reciprocal without the minus
The negative reciprocal of 2 is −1/2, not 1/2 and not −2.
Reading the radius from the wrong form
In x² + y² + dx + ey + f = 0 the radius is not √f. Completing the square moves f and adds (d/2)² and (e/2)².
Worked example
This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.
The point P lies on the ray that starts at the origin and passes through (4, 3). The distance from P to (8, 1) is 5. What is the greater possible distance from P to the origin?
Answer: E. 10
Worked solution. Write P = (4t, 3t) with t > 0. Then (4t − 8)2 + (3t − 1)2 = 25 expands to 25t2 − 70t + 40 = 0, or 5t2 − 14t + 8 = 0, so (5t − 4)(t − 2) = 0. Thus t = 2 or t = 4/5. The distances from the origin are 5t, namely 10 and 4, and the greater is 10.
Why the other options look right. A is the distance for the nearer point, where t = 4/5. B reports the given distance 5 from (8, 1) instead of the distance from P to the origin. C finds the further point (8, 6) but reports its y-coordinate 6 instead of its distance from the origin. D finds the further point (8, 6) but reports its x-coordinate 8 instead of its distance from the origin.
Paper 1 questions
Paper 1 is Applications of Mathematical Knowledge. Calculators are not allowed.
Q1. The point P lies on the ray that starts at the origin and passes through (4, 3). The distance from P to (8, 1) is 5. What is the greater possible distance from P to the origin?
Free · worked solution included
Answer and worked solution
Answer: E. 10
Worked solution. Write P = (4t, 3t) with t > 0. Then (4t − 8)2 + (3t − 1)2 = 25 expands to 25t2 − 70t + 40 = 0, or 5t2 − 14t + 8 = 0, so (5t − 4)(t − 2) = 0. Thus t = 2 or t = 4/5. The distances from the origin are 5t, namely 10 and 4, and the greater is 10.
Why the other options look right. A is the distance for the nearer point, where t = 4/5. B reports the given distance 5 from (8, 1) instead of the distance from P to the origin. C finds the further point (8, 6) but reports its y-coordinate 6 instead of its distance from the origin. D finds the further point (8, 6) but reports its x-coordinate 8 instead of its distance from the origin.
Q2. A circle has a diameter with endpoints (−1, 4) and (5, −2). What is its radius?
Free · worked solution included
Answer and worked solution
Answer: B. 3√2
Worked solution. The change in x is 6 and the change in y is −6, so the diameter has length √(62 + 62) = √72 = 6√2. The radius is half of that length, which is 3√2.
Why the other options look right. A takes half of the horizontal change 6 and ignores the vertical change. C adds the changes instead of using Pythagoras, taking the diameter as 6 + 6 = 12 and so the radius as 6. D stops at the length of the diameter, 6√2, and forgets to halve it to get the radius. E finds r² = 3² + 3² = 18 and then halves it instead of taking the square root, giving 9.
Q3. A triangle has vertices (0, 0), (8, 2) and (2, 6). What is its area?
Free · worked solution included
Answer and worked solution
Answer: C. 22
Worked solution. One vertex is the origin, so the area is half the absolute value of the cross product of the other two position vectors: (1/2)|8 × 6 − 2 × 2| = (1/2)|48 − 4| = (1/2) × 44 = 22.
Why the other options look right. A multiplies each point's own coordinates, giving (1/2)|8 × 2 − 2 × 6| = 2. B is (1/2) × 8 × 6 = 24, with the subtracted term 2 × 2 omitted. D drops the factor 1/2 and reports 44. E takes the area of the enclosing rectangle, 8 × 6 = 48, instead of the triangle.
Q4. P is (1, 0), Q is (3, 2) and R is (6, 5). The vector from Q to R is k times the vector from P to Q. What is k?
Free · worked solution included
Answer and worked solution
Answer: B. 3/2
Worked solution. The vector PQ is (2, 2) and the vector QR is (3, 3). Then (3, 3) = (3/2) × (2, 2), and both components give the same scale factor. Therefore k = 3/2.
Why the other options look right. A is the slope of PQ, which is 2/2 = 1, not the scale factor. C is the x-component of QR, with no division by the x-component of PQ. D uses the whole segment from P to R: PR = (5, 5) = (5/2) × PQ. E is the reciprocal, from writing PQ = (2/3) × QR.
Q5. A circle has equation (3x − 6)2 + (3y + 3)2 = 90. Find its radius.
Free · worked solution included
Answer and worked solution
Answer: E. √10
Worked solution. Factor 3 from each bracket: (3(x − 2))2 + (3(y + 1))2 = 90, so 9((x − 2)2 + (y + 1)2) = 90. Dividing by 9 gives (x − 2)2 + (y + 1)2 = 10. The radius is √10.
Why the other options look right. A takes the square root of 90 and does not divide the equation by 9, giving 3√10. B reports the squared radius 10. C divides 90 by 3 rather than by 9, leaving radius √30. D mistakes the factor 3 taken out of each bracket for the radius.
Q6. Find the acute angle between the lines y = 2x − 1 and y = −3x + 4.
Free · worked solution included
Answer and worked solution
Answer: C. π/4
Worked solution. The gradient of a line is the tangent of the angle it makes with the positive x-axis. For gradients m1 = 2 and m2 = −3, the tangent of the angle θ between the lines satisfies tan θ = |(m1 − m2)/(1 + m1 m2)| = |(2 − (−3))/(1 + 2 × (−3))| = |5/(1 − 6)| = 1. The acute angle with tangent 1 is π/4.
Why the other options look right. A uses only |m1 − m2| = 5 and drops the denominator, giving arctan(5). B treats the lines as perpendicular because the product of the gradients is negative, although perpendicular lines need m1 m2 = −1, not −6. D puts a minus into the denominator, using 1 − m1 m2 = 7, so tan θ = 5/7. E drops the absolute value: (m1 − m2)/(1 + m1 m2) = 5/(−5) = −1 gives the obtuse angle 3π/4, which is not the acute angle asked for.
Q7. The circle x2 + y2 + 4x − 6y = k passes through the point (1, 5). Find its radius.
Free · worked solution included
Answer and worked solution
Answer: C. √13
Worked solution. Complete the square: (x + 2)2 + (y − 3)2 = k + 13. The point (1, 5) gives 32 + 22 = k + 13, so 13 = k + 13 and k = 0. The squared radius is 13, and the radius is √13.
Why the other options look right. A substitutes (1, 5) into x2 + y2 only, getting 1 + 25 = 26, and takes 26 as the squared radius, giving √26. B reports the squared radius 13 instead of its square root. D adds the horizontal and vertical distances from the centre (−2, 3) to (1, 5), 3 + 2 = 5, instead of combining them with Pythagoras. E takes the centre as (2, 3), with the wrong sign for the x-coordinate, and computes its distance to (1, 5): √(12 + 22) = √5.
Q8. A circle has centre (6, 8) and radius 5. What is the greatest possible distance from the origin to a point on the circle?
Free · worked solution included
Answer and worked solution
Answer: D. 15
Worked solution. The distance from the origin to the centre is √(62 + 82) = √(36 + 64) = √100 = 10. The greatest distance from the origin to a point on the circle is that distance plus the radius, which is 10 + 5 = 15. The point lies on the ray from the origin through the centre.
Why the other options look right. A gives the least distance instead of the greatest, 10 − 5 = 5, subtracting the radius rather than adding it. B gives the length of the tangent from the origin to the circle, √(102 − 52) = 5√3, instead of the greatest distance. C is the distance from the origin to the centre, with the radius not added. E is 6 + 8 + 5, adding the coordinates of the centre instead of using the distance √(62 + 82).
Q9. The line 2x − y = 5 is translated by the vector (3, −2). Which equation describes the image of the line?
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Answer
Answer: A. 2x − y = 13
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Q10. What is the equation of the circle that passes through the points (0, 1), (4, 1) and (2, 3)?
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Answer
Answer: D. x2 + y2 − 4x − 2y + 1 = 0
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Q11. A point P moves in the plane so that its distance from (8, 0) is three times its distance from the origin. What is the equation of the locus of P?
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Answer
Answer: A. (x + 1)2 + y2 = 9
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Q12. The Manhattan distance from (x, y) to the origin is |x| + |y|. How many of the following statements are true? I: The set |x| + |y| = 4 is a square of side length 4√2. II: The set |x| + |y| = 4 is a circle of radius 4. III: The set |x| + |y| ≤ 4 has area 32. IV: The set |x| + |y| ≤ 4 has area 16.
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Answer
Answer: C. 2
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Q13. A circle has centre at the origin and radius r. A point P moves on the line x + y = 6. From P, the two tangents to the circle meet at angle θ. The largest possible value of θ is 90°. What is r?
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Answer
Answer: C. 3
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Q14. How many real values of m make the line y = mx + 4 tangent to the circle x2 + y2 = 4?
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Answer
Answer: C. 2
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Q15. The real numbers a and b satisfy (a + 1)2 + (b − 2)2 = 1. What is the minimum possible value of (x − a)2 + (ln x − b)2 as a and b vary subject to this condition and x ranges over the positive reals?
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Answer
Answer: C. 9 − 4√2
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Q16. The line L has equation y = 12 − 3x. A second line is perpendicular to L and passes through (−2, 0). What is the area of the region enclosed by the two lines and the x-axis?
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Answer
Answer: A. 27/5
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Q17. Two tangents are drawn from the origin to the circle x2 + y2 − 6x − 4y + 9 = 0. One of them is the x-axis. What is the gradient of the other tangent?
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Answer
Answer: E. 12/5
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Q18. Find the shortest distance between the circles (x − 1)2 + (y + 2)2 = 8 and (x − 7)2 + (y − 6)2 = 2.
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Answer
Answer: D. 10 − 3√2
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Q19. The circles (x + 2)² + (y − 1)² = 36 and (x − 7)² + (y − 13)² = r², with r > 0, have exactly one point in common. What is the difference between the two possible values of r?
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Answer
Answer: C. 12
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Q20. Find the shortest distance between the curve y = x² + 6 and the line y = 4x − 1.
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Answer
Answer: B. 3√17 / 17
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Q21. The circle C1 has equation (x + 1)2 + (y − 2)2 = 4. The circle C2 has equation (x − 5)2 + (y − 2)2 = 4. The line L is a tangent to both circles and has positive gradient. The acute angle between L and the x-axis is θ. What is tan θ?
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Answer
Answer: E. 2√5/5
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Q22. Two circles have the same radius. One has centre (−1, 4) and the other has centre (5, 2). They meet at two distinct points. What is the equation of the line through those two points?
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Answer
Answer: E. 3x − y = 3
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Q23. The line y = 3x + 1 meets the curve y = x2 + bx + c at exactly one point. The line y = −x + 5 also meets that curve at exactly one point. What is b − c?
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Answer
Answer: B. −6
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Q24. The curve x = y2 − 2y + 4 is rotated 90° clockwise about the point P(1, 1). What is the equation of the image?
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Answer
Answer: A. y = −x2 + 2x − 2
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Q25. Find the length of the curve log10(x) + log10(8 − x) = 2 log10(y − 1).
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Answer
Answer: D. 4π
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Q26. Find the complete set of real a for which x2 − 2ax + y2 − 2y + 3a + 1 = 0 is a circle.
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Answer
Answer: B. a < 0 or a > 3
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Q27. Point P lies on the circle (x − 1)² + (y − 2)² = 9. Point Q lies on the circle (x − 5)² + (y + 2)² = 9. What is the greatest possible length of PQ?
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Answer
Answer: C. 6 + 4√2
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Q28. A right-angled triangle has vertices at (0, 2), (6, 0) and (4, k). Find the sum of all possible values of k.
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Answer
Answer: E. 10
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Q29. For how many positive integers n does the circle x² + y² − 2nx − 2ny + n² = 0 have at least one point in common with the circle x² + y² = 100?
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Answer
Answer: A. 20
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Q30. The perpendicular bisector of the segment joining (1, 4) and (5, −2) meets the x-axis at the point with x-coordinate
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Answer
Answer: A. 3/2
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Q31. What is the midpoint of the points (2, −4) and (8, 6)?
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Answer and worked solution
Answer: A. (5, 1)
Worked solution. The midpoint is ((2 + 8) / 2, (−4 + 6) / 2) = (10 / 2, 2 / 2) = (5, 1).
Why the other options look right. B averages the absolute values of the y-coordinates, (4 + 6) / 2 = 5, and drops the sign of −4. C uses the difference 8 − 2 as the x-coordinate. D halves the differences instead of the sums: ((8 − 2) / 2, (6 − (−4)) / 2) = (3, 5). E adds the coordinates and does not divide by 2.
Q32. The circle x² + y² − 2y = 8 has a tangent that passes through (0, 6) and meets the positive x-axis. What is the x-coordinate of the point where this tangent meets the x-axis?
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Answer
Answer: C. 9/2
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Q33. The point (2a, a), where a ≠ 0, is reflected in the line y = 2x. Which ordered pair is the image?
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Answer
Answer: D. (−2a/5, 11a/5)
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Q34. Chords of the circle x2 + y2 = 16 are drawn through the point P(2, 0). Which equation is satisfied by the midpoint M of every such chord?
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Answer
Answer: B. x2 + y2 − 2x = 0
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Q35. A circle has centre (1, −2) and radius 13. Which of these points lies on the circle?
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Answer
Answer: E. (6, 10)
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Q36. A circle has centre (−1, 2) and radius 5. Which point lies on the circle?
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Answer
Answer: C. (2, 6)
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Q37. A circle has centre (2, 1) and radius 5√2. Which point lies on the circle?
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Answer
Answer: B. (3, −6)
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Q38. A circle has centre (−1, 3) and radius 17. Which of these points lies on the circle?
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Answer
Answer: C. (7, 18)
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Q39. A circle has centre (2, −1) and radius √85. Which of these points lies on it?
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Answer
Answer: B. (8, 6)
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Q40. Which of these points lies on the circle x² + y² − 2x − 4y = 69?
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Answer
Answer: D. (6, 9)
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Q41. A circle passes through the points (0, 0), (12, 0) and (0, 10). Which of these points also lies on the circle?
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Answer
Answer: D. (1, 11)
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Q42. The line through the points (3k, k − 2) and (k + 4, 2k + 1) is perpendicular to the line 2x + y − 1 = 0. What is the value of k?
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Answer
Answer: A. −1/2
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Q43. What is the shortest distance from the point (7, 0) to the curve y = x2 + 2?
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Answer
Answer: C. 3√5
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Q44. A tangent to the circle x2 + y2 − 6y = 7 passes through (0, 11) and meets the positive x-axis. What is the x-coordinate of that meeting point?
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Answer
Answer: E. 11√3 / 3
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Q45. What is the reflection of the point (2, 6) in the line y = 2x?
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Answer
Answer: D. (18/5, 26/5)
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Q46. The line y = 3x + c meets the circle x2 + y2 = 25 in two distinct points P and Q whenever −5√10 < c < 5√10. M is the midpoint of PQ. Which equation describes the path traced by M as c varies over that interval?
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Answer
Answer: B. y = −x/3
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Q47. At the point (p, q) on the curve y = 7 − x2, the normal line has positive gradient and crosses the y-axis at (0, 2). Find p.
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Answer
Answer: B. (3√2)/2
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Q48. The parabola y = x2 + 4 does not meet the line y = x + 1. What is the least distance from a point on the parabola to a point on the line?
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Answer
Answer: C. (11√2)/8
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Q49. At the point on y = 12/x where x = 2, the normal crosses the coordinate axes at P and Q. How long is the segment PQ?
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Answer
Answer: B. (16√10)/3
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Q50. The straight line y = mx + 5 passes through both (3, log base 2 of p) and (log base 2 of p, 5). The constant p can take two positive values. Find the positive difference of those two values.
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Answer
Answer: D. 31
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Q51. The point P lies on the segment from O(0, 0) to Q(9, 6), with OP:PQ = 2:1. The point P is then reflected in the line y = x. What are the coordinates of its image?
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Answer
Answer: A. (4, 6)
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Q52. P is a variable point of the circle x2 + y2 − 2x − 4y = 11, and O is the origin. By how much does the greatest possible length OP exceed the least possible length OP?
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Answer
Answer: E. 2√5
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Q53. A triangle has vertices (1, 1), (7, 1) and (1, 9). One circle passes through all three vertices, and another is tangent to all three sides. Find the area of the region that lies inside the first circle and outside the second.
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Answer
Answer: C. 21π
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Q54. Find the area of the region enclosed by the curve y = √(4 − x2) and the line y = √3 (2 − x).
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Answer: C. 2π/3 − √3
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Q55. For which value of p are the lines (1 + √2) y = p x + 6 and y = (3 − 2√2) x + 5 perpendicular?
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Answer
Answer: E. −7 − 5√2
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Q56. Among all points (x, y) inside or on the circle (x − 2)2 + (y + 1)2 = 9, what is the greatest value of 3x + 4y?
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Answer
Answer: C. 17
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Q57. Point P lies on (x − 2)² + (y − 3)² = 1 and point Q lies on (x + 2)² + (y − 6)² = s, where 0 < s < 40. The values of s for which the shortest possible distance PQ is strictly less than 1 form an interval. What is its length?
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Answer
Answer: B. 31
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Q58. The points (4, 0), (1, 3) and (1, −3) lie on a circle with centre (a, 0) and radius r. What is r − a?
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Answer
Answer: B. 2
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Q59. A circle of radius r, where r > 1, has centre (0, c). It touches the curve y = √(x² + 1) at two points whose x-coordinates are non-zero. What is c in terms of r?
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Answer
Answer: D. √(2r² + 2)
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Paper 2 questions
Paper 2 is Mathematical Reasoning. It can test this same topic. Argument, proof, and identifying errors are the topics that appear on Paper 2 only.
Q1. The circle x2 + y2 − 12x + m = 0 is tangent to the circle x2 + y2 = 4. Which values can m take?
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Answer
Answer: A. 20 or −28
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Q2. PQRS is a rectangle. The coordinates of P and Q are (0, 2) and (3, 6) respectively. The perpendicular to PQ at Q meets the x-axis at R. What is the area of PQRS?
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Answer
Answer: D. 50
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Q3. A path starts at (0, 0). Its first step is 2 units in the positive y direction. After every step it turns 90° anticlockwise. The step lengths are 2, 2, 4, 4, 6, 6, and so on, with each positive even length used twice. Which of the following points is not on the path?
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Answer
Answer: C. (7, 8)
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Q4. A(−1, 2) and C(5, 6) are opposite vertices of the square ABCD. What is the equation of the straight line through B and D?
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Answer
Answer: D. y = −(3/2)x + 7
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Q5. A circle has equation (x − 8)2 + (y − 3)2 = 4. A square has vertices (1, 0), (1, 2), (3, 2) and (3, 0). A straight line bisects both the area of the circle and the area of the square. What is the x-coordinate of the point where this line meets the x-axis?
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Answer
Answer: C. −1
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Q6. A circle has equation x2 + ax + y2 + by + c = 0, where a, b and c are non-zero real constants. Which condition is necessary and sufficient for the circle to be tangent to the line y = x?
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Answer: A. (a + b)2 = 8c
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Q7. The equation x2 + y2 + 2dx + 2ey + c = 0, where d, e and c are real, represents a circle. Which information is enough, by itself, to decide whether the origin lies inside, on or outside the circle?
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Answer: B. the value of c only
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Q8. The diagonals of kite PQRS meet at right angles at O, and the kite is symmetric about the diagonal QS. OP = OR = 3, OQ = 4 and OS = s, where s > 0. Angle QPS is 120°. What is s?
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Answer: E. (48 + 25√3)/13
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Q9. Three lines have equations 2ax + by + c = 0, 2bx + cy + a = 0 and 2cx + ay + b = 0, where a, b and c are nonzero real numbers. Which statement is correct?
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Answer: C. If two of the lines are perpendicular, then the third is parallel to y = 8x.
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Q10. What is the radius of the circle 2x2 + 2y2 − 12x + 4y + 7 = 0?
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Answer: B. √(13/2)
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Q11. The circle with centre (t, 1) and radius t touches the circle with centre (−t, −2t) and radius 2t at exactly one point. Which positive value of t makes this happen?
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Answer: D. 2 + √5
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Q12. What is the shortest distance from a point of the curve y = 1 − x2 to the line 2x + y = 9?
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Answer: E. (7√5)/5
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Q13. Circle C1 has centre (1, 2) and radius 10. Circle C2 has centre (1, 8) and radius r, where r > 0. The circles touch, and one lies inside the other. Which values of r are possible?
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Answer: C. r = 4 or r = 16
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Q14. What is the least distance between the curve y = x² + 2 and the line y = 4x − 3?
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Answer: E. √17/17
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Q15. The line 3x − 2y = 12 is reflected in the line y = x. Which equation describes the reflected line?
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Answer: C. −2x + 3y = 12
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Q16. A circle has centre (0, 0) and radius 11. A second circle has centre (3, 0) and radius r, where r > 0. The two circles touch internally. Which statement gives the possible values of r?
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Answer: A. r = 8 or r = 14
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Q17. What is the shortest distance between the curve y = x² + 3x and the line y = x − 2?
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Answer: D. √2/2
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Q18. The points O(0, 0), P(6, 0) and Q(0, 3) are the vertices of a triangle. What is the length of the perpendicular from O to the line PQ?
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Answer: E. 6√5/5
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Q19. Circle P has centre (0, 0) and radius 9. Circle Q has centre (5, 0) and radius r, where r > 0. The circles touch, and one lies inside the other. Which values of r are possible?
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Answer: E. r = 4 or r = 14
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Q20. What is the shortest distance between the curve y = x² + 5 and the line y = 2x?
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Answer: B. (4√5)/5
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Q21. P is the origin, Q is (6, 0) and R is (0, 9). What is the perpendicular distance from P to the line through Q and R?
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Answer: B. 18√13/13
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Q22. Circle C1 has centre (0, 0) and radius 13. Circle C2 has centre (5, 0) and radius k, where k > 0. The circles touch, and one of them lies inside the other. Which values of k are possible?
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Answer: E. k = 8 or k = 18
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Q23. The triangle PQR has vertices P(0, 0), Q(7, 0) and R(0, 24). What is the distance from P to the line through Q and R?
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Answer: E. 168/25
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Q24. A circle has centre (0, 0) and radius 12. Another circle has centre (5, 0) and radius r, with r > 0. The circles touch, and one lies inside the other. Which values of r are possible?
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Answer: A. r = 7 or r = 17
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Q25. Find the shortest distance between the curve y = x² + 2 and the line y = x.
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Answer: E. (7√2)/8
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Q26. The points P(0, 0), Q(4, 0) and R(0, 7) are the vertices of a triangle. Find the perpendicular distance from P to the line QR.
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Answer: E. 28√65/65
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Q27. Two circles of radius 5 have centres (0, 0) and (6, 0). What is the length of their common chord?
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Answer: D. 8
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Q28. What is the shortest distance between the parabola y = x² + 1 and the line y = x?
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Answer: D. 3√2/8
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Q29. In the triangle with vertices P(0, 0), Q(20, 0) and R(0, 21), find the perpendicular distance from P to the line through Q and R.
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Answer: D. 420/29
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Q30. Circle C1 has centre (0, 0) and radius 8. Circle C2 has centre (3, 0) and radius r, where r > 0. The circles touch internally. Which values of r are possible?
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Answer: E. r = 5 or r = 11
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Q31. What is the shortest distance between the parabola y = x² + 5 and the line y = x?
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Answer: B. 19 √2 / 8
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Q32. The points P(0, 0), Q(12, 0) and R(0, 35) form a triangle. What is the length of the perpendicular from P to the line QR?
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Answer: B. 420/37
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Q33. For which values of k does the line y = x + 1 cross or touch the curve y = x2 + kx + 4?
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Answer: D. k ≤ 1 − 2√3 or k ≥ 1 + 2√3
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Q34. The circle x2 + y2 − 6x − 8y + k = 0 meets the y-axis at two distinct points and does not meet the x-axis. Which of the following describes all the possible values of k?
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Answer: E. 9 < k < 16
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Q35. The circles x2 + y2 − 8x − 6y + 16 = 0 and x2 + y2 = r2, with r > 0, touch each other externally. What is r?
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Answer: B. 2
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Q36. The point (3, 1) is reflected in the line y = x, and the image is then reflected in the line x + y = 6. What are the coordinates of the final image?
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Answer: C. (3, 5)
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Q37. The two tangents from the point T(4, 6) to the circle x2 + y2 − 2x − 4y = 4 touch the circle at P and Q. What is the length of PQ?
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Answer: E. 24/5
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Q38. The circles (x − r)2 + (y − 1)2 = r2 and (x + r)2 + (y + 1)2 = 4 r2 touch exactly once, and r is positive. What is r?
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Answer: B. 2√5 / 5
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Q39. What is the shortest distance between the parabola y = 2 − x2 and the line x + y = 5?
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Answer: A. 11√2 / 8
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