Conic Sections – Class 11 Mathematics notes, important questions and MCQs

Chapter overview

Circles, parabolas, ellipses and hyperbolas are curves formed by cutting a double cone with a plane. Each has a standard equation and key parts like focus and directrix.

Circle and parabola

• Circle: (x − h)² + (y − k)² = r²; general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f), r = √(g² + f² − c).
• Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a.
• Other parabolas: y² = −4ax, x² = 4ay, x² = −4ay.

Ellipse and hyperbola

• 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.
• Eccentricity: circle 0, parabola 1.

Key terms

Eccentricity

ratio that measures how much a conic deviates from a circle

Latus rectum

chord through the focus perpendicular to the axis

Directrix

a fixed line such that the ratio of a point's distance from the focus to its distance from this line equals the eccentricity

Formulas

• Ellipse c² = a² − b²
• Hyperbola c² = a² + b²
• Parabola latus rectum = 4a

Important questions with answers

Quick revision

• Ellipse e < 1.
• Hyperbola e > 1.
• Parabola e = 1.

Exam tips

• For y² = 4ax, the coefficient of x divided by 4 gives a.
• First write the equation in standard form, compare to find a and b, then compute c, e, foci, vertices and latus rectum in that order.

Practice MCQs

1. Eccentricity of a parabola is

   1. 0
   2. less than 1
   3. 1
   4. greater than 1
   Show answer

   Answer: (c) 1. e = 1.

2. Centre of the circle (x − 2)² + (y + 3)² = 16 is

   1. (2, 3)
   2. (2, −3)
   3. (−2, 3)
   4. (4, 16)
   Show answer

   Answer: (b) (2, −3). (h, k).

3. Focus of y² = 12x is

   1. (12, 0)
   2. (3, 0)
   3. (0, 3)
   4. (4, 0)
   Show answer

   Answer: (b) (3, 0). 4a = 12.

4. Radius of x² + y² − 4x − 6y − 12 = 0 is

   1. 4
   2. 5
   3. 25
   4. √12
   Show answer

   Answer: (b) 5. √(4 + 9 + 12).

5. Equation of the circle with centre (0, 0) passing through (3, 4) is

   1. x² + y² = 5
   2. x² + y² = 7
   3. x² + y² = 25
   4. x² + y² = 49
   Show answer

   Answer: (c) x² + y² = 25. r = 5.

6. Eccentricity of x²/25 + y²/16 = 1 is

   1. 5/3
   2. 4/5
   3. 3/5
   4. 3/4
   Show answer

   Answer: (c) 3/5. c = 3, a = 5.

7. Length of latus rectum of x²/9 − y²/16 = 1 is

   1. 32/3
   2. 9/2
   3. 16/3
   4. 8
   Show answer

   Answer: (a) 32/3. 2b²/a = 32/3.

8. Foci of x²/16 + y²/9 = 1 are

   1. (±5, 0)
   2. (±√7, 0)
   3. (0, ±√7)
   4. (±4, 0)
   Show answer

   Answer: (b) (±√7, 0). c² = 16 − 9.