Question

At the point of intersection of the rectangular hyperbola

xy = c² and the parabola y² = 4ax tangents to the rectangular hyperbola and the parabola make an angle q and f respectively with x- axis, then–

• A. q = tan^–1 (–2 tan f)
• B. q = $\frac{1}{2}$tan^–1 (–tan f)
• C. f = tan^–1 (–2 tan q)
• D. f = $\frac{1}{2}$tan^–1 (– tan q)

Answer 1

Answer: (A)

q = tan^–1 (–2 tan f)

Answer 2

Let (x₁, y₁) be point of intersection

Ž y₁² = 4ax₁, x₁y₁ = c²

y² = 4ax xy = c²

$m_{T_{(x_{1},y_{1})}}$= $\frac{2a}{y_{1}}$= tan f

$M_{T_{(x_{1},y_{1})}}$= –$\frac{y_{1}}{x_{1}}$= tan q

So $\frac{\tan\theta}{\tan\varphi}$= $\frac{- y_{1}^{2}}{2ax_{1}}$= – $\frac{4ax_{1}}{2ax_{1}}$= – 2

So q = tan^–1 (– 2 tan f)