Manipal Entrance Test 2013 Math Paper

Let the coordinates of P be (h,k) Let the equation of a tangent from P(h,K) to the circle x2+y2=a2 be y=mx+a√1+m2
since, P(h,K) lies on y=mx+a√1+m2 therefore, k=mh+a√1+m2 =(k−mh)2=a2(1+m2) thisis a quadratic equation in m. Let the two roots be m1 and m2, then
m1+m2=

2hK

K2−a2

But tan α=m1,tan‌β=m2 and it is given that
cot‌α+cot‌β=0
∴

1

m1

+

1

m2

=0
m1+m2=0

2hK

K2−a2

=0
hK=0
Hence, the locus of (h, K) is xy = 0.