Conic Sections — Class 11 Mathematics Worksheet | AAI ATC

Before you begin

Formulas you will need

Key formulas Circle: x² + y² + 2gx + 2fy + c = 0 ⇒ centre (−g, −f), radius √(g² + f² − c)
Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, e = 1
Ellipse x²/a² + y²/b² = 1 (a > b): c² = a² − b², e = c/a < 1, latus rectum 2b²/a
Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a > 1, latus rectum 2b²/a
Common chord or tangent of two circles S₁ and S₂: S₁ − S₂ = 0

How to use this worksheet. For a circle in general form the centre is (−g, −f) — the signs flip. For an ellipse, the major axis has length 2a, so halve it before using it. And for two circles, subtracting one equation from the other kills the squared terms and hands you the common tangent in one line.

Section A · Level 1

Standard forms and eccentricity

Ten questions. None should take more than twenty seconds.

Q1

The equation of a circle with centre at the origin and radius r is:

Every point at distance r from the origin satisfies x² + y² = r².

Q2

For the circle x² + y² + 2gx + 2fy + c = 0, the centre is:

Completing the square shifts the centre to (−g, −f). The signs are opposite to those in the equation, which is the commonest slip.

Q3

For the same circle, the radius is:

The radius is √(g² + f² − c). If that quantity is negative, no real circle exists.

Q4

The eccentricity of a parabola is:

A parabola has eccentricity exactly 1. A circle has 0, an ellipse less than 1 and a hyperbola more than 1.

Q5

The eccentricity of an ellipse is always:

Since c is smaller than a for an ellipse, the ratio c/a lies strictly between 0 and 1.

Q6

The eccentricity of a hyperbola is always:

For a hyperbola c² = a² + b², so c exceeds a and the eccentricity is greater than 1.

Q7

For the parabola y² = 4ax, the focus is at:

The parabola opens along the positive x-axis with its focus at (a, 0) and its directrix at x = −a.

Q8

For the ellipse x²/a² + y²/b² = 1 with a greater than b, the relation between a, b and c is:

For an ellipse the focal distance satisfies c² = a² − b². The plus sign belongs to the hyperbola.

Q9

The length of the latus rectum of the parabola y² = 4ax is:

The latus rectum is the focal chord perpendicular to the axis, and its length is 4a.

Q10

The line obtained from S₁ − S₂ = 0 for two circles is called their:

Subtracting the two equations removes the squared terms and leaves a straight line, the radical axis. When the circles touch, that line is their common tangent.

Section B · Level 2

Centres, foci and latus rectum

Ten questions. Complete the square before reading off the centre.

Q11

The radius of the circle x² + y² = 25 is:

Comparing with x² + y² = r² gives r² = 25, so r = 5.

Q12

The centre and radius of the circle x² + y² − 6x + 4y − 12 = 0 are:

Here g = −3 and f = 2, so the centre is (3, −2) and the radius is √(9 + 4 + 12) = 5.

Q13

The equation of the circle with centre (2, −3) and radius 4 is:

The standard form is (x − h)² + (y − k)² = r², so the signs inside the brackets are opposite to the coordinates of the centre.

Q14

For the parabola y² = 12x, the focus is at:

Comparing with y² = 4ax gives 4a = 12, so a = 3 and the focus is (3, 0).

Q15

For the same parabola y² = 12x, the directrix is:

The directrix of y² = 4ax is the line x = −a, which is x = −3 here.

Q16

The length of the latus rectum of the parabola y² = 12x is:

The latus rectum is 4a, and 4a is the coefficient of x, namely 12.

Q17

For the ellipse x²/25 + y²/9 = 1, the eccentricity is:

Here a = 5 and b = 3, so c = √(25 − 9) = 4 and e = c/a = 4/5.

Q18

For the ellipse x²/25 + y²/9 = 1, the length of the latus rectum is:

The latus rectum of an ellipse is 2b²/a = 2 × 9 ∕ 5 = 18/5.

Q19

For the hyperbola x²/16 − y²/9 = 1, the eccentricity is:

For a hyperbola c² = a² + b² = 16 + 9 = 25, so c = 5 and e = 5/4, which is greater than 1 as it must be.

Q20

For the ellipse x²/9 + y²/4 = 1, the eccentricity is:

With a = 3 and b = 2, c = √(9 − 4) = √5, so e = √5/3.

Section C · Level 3

Definitions, asymptotes and degenerate cases

Seven questions. Check whether the conic is real before answering.

Q21

A conic section is obtained by intersecting a double cone with:

Circle, ellipse, parabola and hyperbola all arise as plane sections of a double right circular cone, which is where the name comes from.

Q22

If the plane cuts the cone parallel to exactly one generator, the section is:

Cutting parallel to a single slant line gives an open curve with eccentricity 1, the parabola.

Q23

The equation x² + y² + 4x + 6y + 20 = 0 represents:

Here g² + f² − c = 4 + 9 − 20 = −7, which is negative. The radius would be imaginary, so no real circle exists.

Q24

For an ellipse, the sum of the distances of any point from the two foci is:

This constant sum, equal to the length of the major axis, is the defining property of an ellipse. For a hyperbola it is the difference that stays constant.

Q25

The asymptotes of the hyperbola x²/a² − y²/b² = 1 are:

Setting the right hand side to zero factorises the equation into the two lines y = ±(b/a)x, which the curve approaches at large distances.

Q26

If the major axis of an ellipse has length 10, then a equals:

The major axis has length 2a, so a is half of it. Confusing the axis length with a is the trap in one of the previous year questions.

Q27

For a circle, the eccentricity is:

A circle is the limiting case of an ellipse in which the two foci coincide, so c = 0 and the eccentricity is 0.

Section D · Previous year questions

Straight from the AAI ATC papers

Three questions from the 2023 papers, recovered by optical character recognition of the scanned answer sheets and independently re-solved with a computer algebra system.

Q28PYQ 2023 · Morning

What is the centre and radius, respectively, of the circle represented by the equation x² − 10x + y² + 8y + 5 = 0?

Comparing with the general form gives 2g = −10 and 2f = 8, so g = −5 and f = 4 and the centre is (−g, −f) = (5, −4). The radius is √(25 + 16 − 5) = √36 = 6. Option 4 quotes 36, which is the square of the radius rather than the radius itself.

Q29PYQ 2023 · Morning

What is the equation of an ellipse whose focus is at (3, 0) and the length of its major axis is 10?

The major axis has length 2a = 10, so a = 5. The focus at (3, 0) gives c = 3, and then b² = a² − c² = 25 − 9 = 16. The equation is x²/25 + y²/16 = 1. Option 2 uses c² in place of b², and option 4 puts the larger denominator under y, which would place the focus on the y-axis instead.

Q30PYQ 2023 · Evening

The common tangent of the two touching circles x² + y² + 6x − 2y + 7 = 0 and x² + y² − 4x + 7y − 9 = 0 is:

For two circles that touch, the common tangent at the point of contact is found by subtracting one equation from the other. The squared terms cancel, leaving (6x − 2y + 7) − (−4x + 7y − 9) = 10x − 9y + 16 = 0. Watching the signs while subtracting is the entire difficulty — the three wrong options are simply sign errors.

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