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A tangent is drawn at any point on the h...

A tangent is drawn at any point on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2)) =1`. If this tangent is intersected by the tangents at the vertices at points P and Q, then which of the following is/are true

A

S,S',P and Q are concyclic

B

PQ is diameter of the circle

C

S,S', P and Q forms rhombus

D

PQ is diagonal of acute angle of the rhombus formed by S,S',P and Q

Text Solution

Verified by Experts

The correct Answer is:
A, B

Any tangent to the hyperbola is `(x sec theta)/(a) -(y tan theta)/(b) =1` Solving this line with the lines `x = +-a`, we get the coordinates of points P and Q as
`(a,b tan.(theta)/(2))` and `(-a,-b cot.(theta)/(2))`
Now slopes of the lines PS and QS are
`m_(PS) =(btan.(theta)/(2))/(a(1-e)), m_(QS) =(-b cot.(theta)/(2))/(-a(1+e))`
`rArr m_(PS).m_(QS) =(-b^(2))/(a^(2)(e^(2)-1)) =-1`
Similarly `m_(PS). m_(QS') =-1`
`rArr` line PQ subtends and angle of `(pi)/(2)` at S and S'
`rArr` points P,Q,S and S' are concyclic.
`rArr PQ` is diameter.
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