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$If the chord of contact of tangents from a point P to the parabola y^{2} = 4 a x touches the parabola x^{2} = 4 b y, then the$ $locus of P is$

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a

xy=ab

b

xy=2ab

c

xy=−2ab

d

xy=−ab

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detailed solution

Correct option is C

The chord of contact of the parabola y2=4ax w.r.t. point Px1,y1 is yy1=2ax+x1-------(1) This line touches the parabola x2=4by . Solving (1) with parabola, we have x2=4b2ay1x+x1 or y1x2−8abx−8abx1=0According to the question, this equation must have equal roots.Therefore, D = 0 or 64a2b2+32abx1y1=0 or x1y1=−2ab or xy=−2abwhich is the required locus.

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