IMO Practice Test: Conic Sections | Class 11 Mathematics

Question 1 of 6 hard

A circle described on any focal chord of a parabola as diameter always touches the:

Axis of symmetry

Tangent at the vertex

Directrix

Latus rectum

Explanation: This is a standard geometric property of parabolas: The circle described on any focal chord as diameter touches the directrix of the parabola.

Question 2 of 6 hard

What is the distance between the directrices of x²/36 + y²/20 = 1?

18

9

36

27

Explanation: a=6, b²=20. e = √(1 - 20/36) = √(16/36) = 4/6 = 2/3. Distance = 2a/e = 12 / (2/3) = 18.

Question 3 of 6 hard

What is the focal distance of a point (x₁, y₁) on the parabola x² = -4ay?

|y₁ - a|

|y₁ + a|

|x₁ + a|

|x₁ - a|

Explanation: For x² = -4ay, focus is (0, -a) and directrix is y = a. By definition, focal distance equals distance to directrix. Distance from (x₁, y₁) to y = a is |y₁ - a|.

Question 4 of 6 hard

The line y=2 is tangent to which circle?

x²+y²−4y=0

x²+y²−2y=0

x²+y²=1

x²+y²+4y=0

Explanation: Circle centre=(0,2), radius=2. Distance to y=2 is 0, not tangent. For x²+y²−4y=0, tangent at top point y=4? Actually line y=2 passes through centre. Hence not tangent. For x²+y²−2y=0, centre=(0,1), radius=1 and distance to y=2 is 1, tangent.

Question 5 of 6 hard

Identify the foci of 16x² + y² = 16.

(±√15, 0)

(0, ±√15)

(0, ±4)

(±4, 0)

Explanation: x²/1 + y²/16 = 1. Major axis is y-axis. b=4, a=1. c = √(16-1) = √15. Foci are (0, ±√15).

Question 6 of 6 hard

Find the directrix of the parabola x² + 4x + 4y + 8 = 0.

y = 0

y = -1

y = 1

y = -2

Explanation: x² + 4x + 4 = -4y - 4 → (x + 2)² = -4(y + 1). Vertex (-2, -1), opens down, 4a = 4 → a = 1. Directrix is y = k + a = -1 + 1 = 0.

IMO Practice Test — Conic Sections