Question

The locus of a point P $(\alpha ,\beta )$ moving under the condition that the line $y=ax+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

• A. an ellipse
• B. a circle
• C. a parabola
• D. a hyperbola

Answer 1

The correct option is D a hyperbola

Tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $y=mx\pm \sqrt{a^2{m}^2-b^2}$

Given that $y=\alpha x+\beta$ is the tangent of hyperbola

$=>m=\alpha$ and $a^2{m}^2-b^2={\beta }^2$

$\Rightarrow a^2{\alpha }^2-b^2={\beta }^2$

Hence required locus is $a^2{x}^2-y^2=b^2$ which is hyperbola.