The locus of a point $P(\alpha, \beta)$ moving under the condition that the line $y=\alpha x+\beta$ is a…
The locus of a point $P(\alpha, \beta)$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is
an ellipse
a circle
a parabola
a hyperbola
Solution
Tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is
$y=m x \pm \sqrt{a^2 m^2-b^2}$
Given that $y=\alpha x+\beta$ is the tangent of hyperbola
$\Rightarrow m=\alpha$ and $a^2 m^2-b^2=\beta^2$
$\therefore \mathrm{a}^2 \alpha^2-\mathrm{b}^2=\beta^2$
Locus is $a^2 x^2-y^2=b^2$ which is hyperbola.