The points on the hyperbola x2 - y2=a2 from where mutually perpendicular tangents can be drawn to the circle x2+y2=a2 is/are
The equation of the tangent to the circle is:
y=mx+a√1+m2
Let P(h,k) lies on the tangent,
k=mh+a√1+m2
k−mh=a√1+m2
Squaring both sides we get:
k2+(mh)2−2kmh=a2(1+m2)
m2(h2−a2)−2mhk+k2−a2=0
This is a quadratic equation in m
Let roots be m1 and m2:
m1m2=−1 ..... (⊥ tangents)
k2−a2(h2−a2)=−1
h2+k2=2a2
So, the points will lie on:
x2+y2=2a2
Now the point of intersection with the hyperbola is :
x2−y2=a2 ........ (1)
x2+y2=2a2 ........ (2)
Adding equations (1) and (2) we get:
x=a√32
Similarly, y=a√2