Let P be a point from where perpendicular tangents are drawn to the circle 2x 2 + 2y 2 – a 2 = 0. Let a line from P perpendicular to OP is drawn which intersect hyperbola x 2 – y 2 = a 2 at Q and R. Find number of all possible positions of P such that product of ordinates of points Q and R is.
(i) 
(ii) a 2 (iii) 
Text Solution
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(i) 4 (ii) 2 (iii) 0
Sol. Clearly P lies on x 2 + y 2 = a 2 and QR is tangent to this circle.
so from x 2 – y 2 = a 2 and x cos θ – ysin θ = a
⇒ (ysin θ + a) 2 – y 2 cos 2 θ = a 2 cos 2 θ
(cos 2 θ – sin 2 θ )y 2 – 2asin θ .y – a 2 sin 2 θ = 0
y 1 y 2 = 
= 
2sin 2 θ = – 3cos 2 θ + 3sin 2 θ
sin 2 θ = 3cos 2 θ ⇒ tan 2 θ = 3
θ = ± 
, π ± 
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