Question

Consider the ellipse x²/25 + y²/9 = 1 with centre C and P is a point on the ellipse with eccentric angle 45°. The normal drawn at P meets the major and the minor axes at G and g, respectively. F is the foot of the perpendicular drawn from the centre C onto the normal at P. The tangent at P meets the major axis at T. M and N are the feet of the perpendiculars drawn from the foci S and Sa onto the tangent at P. Match the items of Column I with those of Column II.

Column I	Column II
A) PF·PG is equal to	(p) 9
(B) PF·Pg is equal to	(q)  16
(C) PG·Pg is equal to	(r)  17
(D) CG·CT is equal to	(s)  15
	(t)  25

Answer 1

See Fig. We have

P = (a cosθ, b sinθ) = (5/√2, 3/√2)

The tangent at P is

 From Eq. (1), we get

T = (5√2, 0)   ...(4)

 The eccentricity e is given by

9 = 25(1 - e²) ⇒ e = 4/5

so that

 S = (ae,0) = (4,0) and S' = (-4,0)

Also note that SM·S'N is always equal to b². In Problem 6 of previous section (Multiple Correct Choice Type Questions), we have proved that PF·Pg = a² so that

PF.pg = a² = 25

Now,

CG.CT = 8√2/5 .5√2 = 16

Finally,

Answer: (A) → (q); (B) → (t); (C) → (p); (D) → (r)