Question

Locus of the middle points of the chords of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = 1 which passes through a fixed point (a, b) is hyperbola whose centre is –

• A. (a, b)
• B. (2a, 2b)
• C. (a/2, b/2)
• D. None these

Answer 1

Answer: (C)

(a/2, b/2)

Answer 2

T = S₁ at mid point (h, k) $\frac{hx}{a^{2}} - \frac{ky}{b^{2}}$ = $\frac{h^{2}}{a^{2}}$ – $\frac{k^{2}}{b^{2}}$

pass (a, b) point

\ $\frac{h\alpha}{a^{2}}$ – $\frac{ky}{b^{2}}$ = $\frac{h^{2}}{a^{2}}$ – $\frac{k^{2}}{b^{2}}$

\ locus is $\frac{x^{2} - \alpha x}{a^{2}}$ – $\frac{y^{2} - \beta y}{b^{2}}$ = 0

Ž $\frac{(x - \alpha/2)^{2}}{a^{2}}$ – $\frac{(y–\beta/2)^{2}}{b^{2}}$ = $\frac{1}{4}$ $\left( \frac{\alpha^{2}}{a^{2}} - \frac{\beta^{2}}{b^{2}} \right)$

= k² (let)

is a hyperbola.

Where centre is (a/2, b/2).