Question

If the tangent at the point (4 cos f , (16/Ö11) sin f) to the ellipse 16x² + 11y² = 256 is also a tangent to the circle x² + y² – 2x = 15, then the value of f is-

• A. ± p/2
• B. ± p/4
• C. ± p/3
• D. ± p/6

Answer 1

Answer: (C)

± p/3

Answer 2

The equation of the tangent at

(4 cos f, (16/Ö11) sin f) to the ellipse

16x² + 11y² = 256.

is 16x (4 cos f) + 11y ((16/Ö11) sin f) = 256

Ž 4x cos f + Ö11y sin f = 16

This touches the circle (x – 1)² + y² = 4², therefore $\left| \frac{4\cos\varphi - 16}{\sqrt{16\cos^{2}\varphi + 11\sin^{2}\varphi}} \right|$ = 4

Ž (cos f – 4)² = 16 cos² f + 11 sin² f

Ž 15cos² f + 11 sin² f + 8 cos f – 16 = 0

Ž 4 cos² f + 8 cos f – 5 = 0

Ž (2 cos f – 1) (2 cos f + 5) = 0

Ž cos f = 1/2 Ž f = ± p/3.