Conic Section

Maths, HSSC-Punjab Second Year. This chapter is split into 65 topics and has 1,604 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Equation of an ellipse from given data 25 questions
2. Tangent to a circle centred at the origin 27 questions
3. Tangent to a general circle at a point 24 questions
4. Key features of the ellipse 23 questions
5. Electron orbits in the Bohr-Sommerfeld model 24 questions
6. Identifying and comparing conics 28 questions
7. Centre and radius from the general form 27 questions
8. Rectangles and distances related to an ellipse 24 questions
9. Circle definition and standard form 23 questions
10. Parabola given as y equal to a quadratic in x 27 questions
11. Hyperbolas with shifted centre 24 questions
12. Tangents to an ellipse: condition and slopes 29 questions
13. Features of vertical hyperbolas (numerical) 25 questions
14. Circle through two points with centre on a line 21 questions
15. Asymptotes and rectangular hyperbola 26 questions
16. Ellipse definition and derivation 22 questions
17. Conic sections and the cone 23 questions
18. Chord of contact and pair of tangents 26 questions
19. Tangent and normal to an ellipse at a point 24 questions
20. Length of a chord of a circle 25 questions
21. Points, parameters and focal distances of a hyperbola 23 questions
22. Hyperbolic navigation (LORAN) 22 questions
23. Comets on parabolic orbits 23 questions
24. Circle through three non-collinear points 28 questions
25. Circle through two points touching a given line 22 questions
26. Circles touching axes and cutting intercepts 21 questions
27. Circle with given end points of a diameter 27 questions
28. Intercepts and axis conditions of a circle 24 questions
29. Elliptical orbits: apogee, perigee and eccentricity 27 questions
30. Condition of tangency for a parabola 25 questions
31. Features of vertical ellipses (numerical) 26 questions
32. Key features of the hyperbola 22 questions
33. Features of horizontal ellipses (numerical) 30 questions
34. Intersection of a line and a circle 23 questions
35. Tangent and normal to a hyperbola at a point 25 questions
36. Locus problems leading to circles 21 questions
37. Normal to a circle 24 questions
38. Tangents from an external point 25 questions
39. Equation of a parabola from given elements 24 questions
40. Eccentricity and axes problems on hyperbolas 28 questions
41. Suspension bridges and parabolic arches 22 questions
42. Tangent and normal to a parabola at a point 25 questions
43. Elements of the standard parabolas 28 questions
44. Elliptical tracks, gardens and arches 25 questions
45. Parametric equations of a circle 29 questions
46. Length of a tangent from a point 25 questions
47. Ellipses with shifted centre 25 questions
48. Parabolic reflectors and flashlights 26 questions
49. Tangents to a hyperbola: condition and slopes 23 questions
50. Hyperbola definition and derivation 23 questions
51. Equation of a hyperbola from given data 22 questions
52. Equation of a circle from centre and radius 26 questions
53. Circle through two points touching a line at one of them 23 questions
54. Position of a point with respect to a circle 26 questions
55. Condition of tangency and tangents of given slope 22 questions
56. Parabola definition and elements 24 questions
57. Centre and radius from the standard equation 24 questions
58. Features of horizontal hyperbolas (numerical) 29 questions
59. Line and parabola intersection 24 questions
60. Points, parameters and focal distances of an ellipse 23 questions
61. General equation of a circle (theorem) 21 questions
62. Latus rectum, parametric points and focal distance 26 questions
63. Assertion-reason and mixed conceptual questions on conics 22 questions
64. Circle with common coefficient of x squared and y squared 25 questions
65. Parabolas with shifted vertex 29 questions

Sample questions: Conic Section

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Which statement about the cutting plane and the cone is correct?

• A. A plane parallel to a generator gives an ellipse, and a plane perpendicular to the axis gives a parabola.
• B. A plane parallel to the axis gives a parabola, and a plane perpendicular to the axis gives a hyperbola.
• C. A plane parallel to a generator gives a hyperbola, and a plane perpendicular to the axis gives an ellipse.
• D. A plane parallel to a generator gives a parabola, and a plane perpendicular to the axis gives a circle.(correct answer)

Answer: D.Perpendicular to the axis gives a circle; parallel to a generator gives a parabola; parallel to the axis through both nappes gives a hyperbola; slightly inclined through one nappe gives an ellipse.

Q2. The tangent at the point on x² + y² − 4x + 6y + 12 = 0 whose ordinate is −2 (Example 10) is:

• A. y − 2 = 0
• B. x + y = 0
• C. x − 2 = 0
• D. y + 2 = 0(correct answer)

Answer: D.Putting y = −2 gives (x − 2)² = 0, so the point of contact is (2, −2); the tangent is 2x − 2y − 2(x + 2) + 3(y − 2) + 12 = 0, i.e. y + 2 = 0. (The book writes x = ±2 for the point, but x = 2 only.)

Q3. The pair of tangents drawn from (10, 0) to x² + y² = 36 has the combined equation:

• A. 36(x + 10)² − 64y² = 0
• B. 64(x − 10)² − 36y² = 0
• C. 36(x − 10)² − 64y² = 0(correct answer)
• D. 36(x − 10)² + 64y² = 0

Answer: C.The line through (10, 0) and (h, k) is at distance 6 from the origin: 100k² = 36(k² + (h − 10)²), giving 64k² = 36(h − 10)²; the combined equation is 36(x − 10)² − 64y² = 0.

Q4. The pair of tangents drawn from (13, 0) to x² + y² = 25 has the combined equation:

• A. 25(x − 13)² − 144y² = 0(correct answer)
• B. 25(x − 13)² + 144y² = 0
• C. 25(x + 13)² − 144y² = 0
• D. 144(x − 13)² − 25y² = 0

Answer: A.The line through (13, 0) and (h, k) is at distance 5 from the origin: 169k² = 25(k² + (h − 13)²), giving 144k² = 25(h − 13)²; the combined equation is 25(x − 13)² − 144y² = 0.

Q5. The tangents from (10, −2) to the circle (x + 3)² + (y + 2)² = 25 touch it at the points:

• A. (−14/13, 34/13) and (−14/13, −86/13)(correct answer)
• B. (34/13, −14/13) and (−86/13, −14/13)
• C. (14/13, −34/13) and (14/13, 86/13)
• D. (−14/13, 34/13) and (−14/13, 86/13)

Answer: A.Each tangent is at distance 5 from the centre; the point of contact is the foot of the perpendicular from the centre (−3, −2) to the tangent: (−14/13, 34/13) and (−14/13, −86/13).

Q6. The tangents from (−10, −4) to the circle (x + 2)² + (y + 3)² = 49 touch it at the points:

• A. (−38/5, −36/5) and (−110/13, 4/13)
• B. (38/5, 36/5) and (110/13, 4/13)
• C. (−38/5, −36/5) and (−110/13, −4/13)(correct answer)
• D. (−36/5, −38/5) and (−4/13, −110/13)

Answer: C.Each tangent is at distance 7 from the centre; the point of contact is the foot of the perpendicular from the centre (−2, −3) to the tangent: (−38/5, −36/5) and (−110/13, −4/13).

Q7. Which is the correct order of the steps in the rule for finding the length of the tangent?

• A. take the square root, substitute the point, make the right side zero
• B. substitute the point, make the right side zero, take the square root, divide by the coefficients
• C. make the right side 1, substitute the point, square the result
• D. make coefficients of x², y² equal to 1, make the right side zero, substitute the point, take the square root(correct answer)

Answer: D.The book's rule: (i) coefficients of x² and y² equal to 1; (ii) right-hand side zero; (iii) substitute the coordinates of the point in the left side; (iv) take the square root.

Q8. Relative to the hyperbola x²/9 − y²/4 = 1, the line y = 2x − 2 is:

• A. external (no real point of intersection)(correct answer)
• B. a tangent
• C. a normal
• D. a secant (two points)

Answer: A.a²m² − b² = 32 and c² = 4; the discriminant is positive when c² > 32, zero when equal and negative when smaller; here c² < 32: the line is external (no real point of intersection).

Q9. For the hyperbola x²/9 − y²/16 = 1 a normal parallel to the line 2x − y = 5 (Exercise 6.5, Q15) cannot exist as a real line. The reason is:

• A. the line 2x − y = 5 is an asymptote
• B. a normal of slope 2 would need y₁/x₁ = −32/9, steeper than the asymptotes (|slope| > 4/3), so no real point of the curve has that ratio(correct answer)
• C. normals always have slope 1
• D. the hyperbola has no normals

Answer: B.The normal slope −a²y₁/(b²x₁) = 2 gives y₁ = −(32/9)x₁; substituting in the curve gives x₁² = −81/55 < 0, impossible. (The book exercise asks for such normals; no real answer exists.)

Q10. Assertion (A): The tangent to a circle at a point is perpendicular to the radius at that point. Reason (R): The normal to a circle always passes through its centre.

• A. Both A and R are true but R is not the correct explanation of A
• B. A is true but R is false
• C. Both A and R are true and R is the correct explanation of A(correct answer)
• D. A is false but R is true

Answer: C.Since the normal at P is the line perpendicular to the tangent through P, it is the radius line and passes through the centre, which explains why the radius is perpendicular to the tangent.