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Question

The points on the hyperbola x2 - y2=a2 from where mutually perpendicular tangents can be drawn to the circle x2+y2=a2 is/are

A
(a32,a2)
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B
(a32,a2)
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C
(a32,a2)
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D
(a32,a2)
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Solution

The correct option is A (a32,a2)

The equation of the tangent to the circle is:

y=mx+a1+m2

Let P(h,k) lies on the tangent,

k=mh+a1+m2

kmh=a1+m2

Squaring both sides we get:

k2+(mh)22kmh=a2(1+m2)

m2(h2a2)2mhk+k2a2=0

This is a quadratic equation in m

Let roots be m1 and m2:

m1m2=1 ..... ( tangents)

k2a2(h2a2)=1

h2+k2=2a2

So, the points will lie on:

x2+y2=2a2

Now the point of intersection with the hyperbola is :

x2y2=a2 ........ (1)

x2+y2=2a2 ........ (2)

Adding equations (1) and (2) we get:

x=a32

Similarly, y=a2


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