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# 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 $\large\frac{x^2}{a^2} -\frac{y^2}{b^2}$$=1$ is

$\begin{array}{1 1}(A)\;a\; hyperbola \\(B)\;a\; parabola \\(C)\;an\; ellipse \\(D)\;a \;circle \end{array}$

If $y=mx+c$ is tangent to hyperbola, then
$c^2=a^2m^2-b^2$
=> $\beta^2=a^2 \alpha^2 -b^2$
$\therefore$Locus of P is $a^2x^2-y^2=b^2$