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Question

The tangents to x2+y2=a2 having inclinations α and β intersect at P. If cotα+cotβ=0, then the locus of P is

A
x+y=0
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B
xy=0
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C
xy=0
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D
None of these
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Solution

The correct option is B xy=0
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+a1+m2

Since, P(h,k) lies on y=mx+a1+m2

therefore, k=mh+a1+m2=(kmh)2=a2(1+m2) this is a quadratic equation in m.

Let the two roots be m1 and m2, then

m1+m2=2hKK2a2

But tanα=m1, tanβ=m2 and it is given that

cotα+cotβ=0

1m1+1m2=0

m1+m2=0

2hKK2a2=0

hK=0

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

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