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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 $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is
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A hyperbola
If $y=m x+c$ is tangent to the hyperbola then $c^2=a^2 m^2-b^2$. Here $\beta^2=a^2 \alpha^2-b^2$. Hence locus of $P(\alpha, \beta)$ is $a^2 x^2-y^2=b^2$, which is a hyperbola.
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