Question

If the sum of the squares of slopes of the normals from a point P to the hyperbola xy = c2 is equal to l (lĪ R+), then the locus of the point P is–

• A. x2 = lc2
• B. y2 = lc2
• C. xy = lc2
• D. x2y2 = lc2

Answer 1

Answer: (A)

x2 = lc2

Answer 2

Equation of normal at any point

$\left( ct,\frac{c}{t} \right)$ is ct4 – xt3 + ty – c = 0 .......(1)

\ Slope of normal = t2

The normal (1) passes through the point P (h, k)

\ ct4 – ht3 + kt – c = 0. If the roots of this equation are ti ; i = 1, 2, 3, 4

then S ti = $\frac{h}{c}$ and S titj = 0,

S titjtk = –$\frac{k}{c}$ and t1t2t3t4 = – 1

Given S ti2 = l Ž (S ti)2 – 2S titj = l

Ž $\frac{h^{2}}{c^{2}}$ – 0 = l Ž h2 = lc2

\ Required locus is x2 = lc2