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Question: If the tangent and the normal to a rectangular hyperbola at a point cut off intercepts a<sub>1</sub>...

If the tangent and the normal to a rectangular hyperbola at a point cut off intercepts a1, a2 on one axis and b1, b2 the other axis then a1a2 + b1b2 is equal to-

A

–1

B

0

C

1

D

2\sqrt{2}

Answer

0

Explanation

Solution

Let the hyperbola be xy = c2

Tangent at any point (ct, c/t)

x dydx\frac{dy}{dx} + y = 0 Ž(dydx)(ct,ct)\left( \frac{dy}{dx} \right)_{\left( ct,\frac{c}{t} \right)} = – c/tct\frac{c/t}{ct} = – 1t2\frac{1}{t^{2}}

Equation of tangent Ž y – ct\frac{c}{t} = –1t2\frac{–1}{t^{2}} (x – ct)

a1 (Intercept on x-axis) ; 0 – ct\frac{c}{t} = –1t2\frac{–1}{t^{2}} (x – ct)

a1 = 2ct ; b1 (Intercept of y-axis) ; b1 = 2c/t

Equation of Normal Ž y – ct\frac{c}{t} = t2 (x – ct)

a2 = c(t4–1)t2\frac{c(t^{4}–1)}{t^{2}}, b2 = –c(t4–1)t\frac{–c(t^{4}–1)}{t}Ž a1 a2 + b1b2 = 0