Question

 A tangent is drawn to the ellipse x²/a² + y²/b² = 1 to  cut the ellipse x²/c² + y²/d² = 1  at P and Q. If the tangents drawn at P and Q to the ellipse

x²/c² + y²/d² = 1  are at right angles, the

(a)  a²/c² + b²/d² = 1

(b)  c²/a² + d²/b² = 1

(c)   a²d² = b²c²

(d)  a²d² + c²d² = 1

Answer 1

Correct option  (a)  a²/c² + b²/d² = 1

Explanation :

See Fig.  Clearly, if R(h, k) is the intersection of tangents at P and Q, then R lies on the director circle and hence

h² + k² = c² + d² .....(1)

Since PQ is the chord of contract of R, its equation is

hx/c² + ky/d² = 1  ...(2)

 However, the line provided in Eq. (2) touches the ellipse

Equations (1) and (3) hold if h² = c² and k² = d². Therefore

a²/c² + b²/d² = 1