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The line l is tangent to the circle S at the point A. B and C are points on l on opposite sides of A and other tangents from B.  C to S intersect at a point P. If B, C vary along l in such a way that the product |AB| . |AC| is constant, then locus of P is

(A) circle 

(B)  a line parallel to BC

(C)  a set of points

(D)  None of these

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Best answer

Correct option (B) a line parallel to BC

See Fig.

∠PBA =  β, ∠PCA =  γ

AB = p1, AC = q1

p1q1 = k2

Let S touch BP and CP at E.

In ΔPEI

Semi-perimeter of ΔBCP is

PH = 2k2r/k2 - r

So, the locus of P is a line parallel to BC

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