If the chord of contact of tangents from a point P to the parabola y2=4 a x touches the parabola x2=4 b y, then the locus of P is

Question

If the chord of contact of tangents from a point P to the parabola y2=4 a x touches the parabola x2=4 b y, then the locus of P is (A) x y=a b (B) x y=2 a b (C) x y=-2 a b (D) x y=-a b

Tardigrade Admin · Accepted Answer

The chord of contact of the parabola y2=4 a x w.r.t. point P(x1, y1) is yy1 = 2a(x + x1) .... (1) This line touches the parabola x2=4 b y. Solving (1) with parabola, we have x2=4 b[(2 a/y1)(x+x1)] or y1 x2-8 a b x-8 a b x1=0 According to the question, this equation must have equal roots. Therefore, D=0 or 64 a2 b2+32 a b x1 y1=0 or x1 y1=-2 a b or x y=-2 a b which is the required locus.

Q. 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

$x y = ab$

$x y = 2 ab$

$x y = - 2 ab$

$x y = - ab$

Q. If the chord of contact of tangents from a point P to the parabola y2=4ax touches the parabola x2=4by, then the locus of P is

xy=ab

xy=2ab

xy=−2ab

xy=−ab

Q. 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

$x y = ab$

$x y = 2 ab$

$x y = - 2 ab$

$x y = - ab$