The locus of a point Pleft( {alpha ,beta | vedclass

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Question

The line $y=m x+c$ is a tangent to the hyperbola

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

if    $c^{2}=a^{2} m^{2}-b^{2}$

The li

Vedclass · Accepted Answer

a hyperbola

Vedclass · Answer

The line $y=m x+c$ is a tangent to the hyperbola

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

if    $c^{2}=a^{2} m^{2}-b^{2}$

The line $y=\alpha x+\beta$ is a tangent to

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

So, $\beta^{2}=a^{2} \alpha^{2}-b^{2}$

$	herefore $ Locus of $(\alpha, \beta)$ is

$y^{2}=a^{2} x^{2}-b^{2}$

$\Rightarrow a^{2} x^{2}-y^{2}=b^{2}$

or $\frac{x^{2}}{\left(\frac{b}{a}ight)^{2}}-\frac{y^{2}}{b^{2}}=1,$ which is hyperbola.

The locus of a point $P\left( {\alpha ,\beta } \right)$ 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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