Question

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

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

Answer 1

The correct option is C a hyperbola

Tangent to the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$y=mx\pm \sqrt{a^2{m}^2-b^2}$
and it is given that ,
$y=\alpha x+\beta$ is the tangent to the hyperbola
$m=\alpha and{a}^2{m}^2-b^2={\beta }^2$
$a^2{\alpha }^2-b^2={\beta }^2$
Locus is $a^2{x}^2-b^2=y^2$ which is hyperbola
Hence, Option C.