The locus of a point P(alpha, beta) moving un | vedclass

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(B) The condition for the line $y = mx + c$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $c^2 = a^2m^2 - b^2$.Given the

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$A$ hyperbola

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(B) The condition for the line $y = mx + c$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $c^2 = a^2m^2 - b^2$.Given the line $y = \alpha x + \beta$,we have $m = \alpha$ and $c = \beta$.Substituting these into the condition,we get $\beta^2 = a^2\alpha^2 - b^2$.Replacing $(\alpha, \beta)$ with $(x, y)$,the locus is $y^2 = a^2x^2 - b^2$,which can be rewritten as $a^2x^2 - y^2 = b^2$.This equation represents a hyperbola.

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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