Question

If normal to hyperbola xy = c² at $\left( ct_{1},\frac{c}{t_{1}} \right)$ meet the curve again at $\left( ct_{2},\frac{c}{t_{2}} \right)$, then:

• A. t₁t₂ = -1
• B. t₂ = -t₁ - $\frac{2}{t_{1}}$
• C. $t_{1}t_{1}^{3} = - 1$
• D. $t_{1}^{3}t_{2} = - 1$

Answer 1

Answer: (D)

$t_{1}^{3}t_{2} = - 1$

Answer 2

Equation of normal at $\left( ct_{1},\frac{c}{t_{1}} \right)$ is

$t_{1}^{3}x - t_{1}y - ct_{1}^{4} + c = 0$.

It passes through $\left( ct_{2},\frac{c}{t_{2}} \right)$

i.e., $t_{1}^{3}.ct_{2} - t_{1}.\frac{c}{t_{2}} - ct_{1}^{4} + c = 0$

⇒ $(t_{1} - t_{2})(t_{1}^{3}t_{2} + 1) = 0$

⇒ $t_{1}^{3}t_{2} = - 1$ (as t₁ ≠ t₂)