Set of values of ‘h’ for which the number of distinct commo…

Maths Conic Sections Mix Single Correct MCQ

Published by: CGP EDU Academic Team

Published on: August 13, 2026

Set of values of ‘h’ for which the number of distinct common normals of

(x – 2) 2 = 4(y – 3) and x 2 + y 2 – 2x – hy – c = 0 (c > 0) is 3, is

(2, ∞ )

(4, ∞ )

(2, 4)

(10, ∞ )

Text Solution

Verified by Experts

The correct answer is:

The equation of any normal

(x – 2) 2 = 4(y – 3) is x – 2 = m (y – 3) – 2m – m 3 If it passes through (1, h/2), then

1 – 2 = m – 2m – m

3 ⇒ 2m 3 + m

(10 – h) – 2 = 0

This equation will give three distinct values of m, if ƒ ′ (m)

= 0 has two distinct roots,

where ƒ(m) = 2m 3 + m (10 – h) – 2

Now ƒ ′ (m) = 6m 2 + (10 – h)

ƒ ′ (m) = 0 ⇒ m ±

The values of m are real and distinct if h > 10 i.e. h ∈ (10, ∞ ).

Hence is correct answer.

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