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

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

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a hyperbola

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(A) The line $y = mx + c$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ if $c^2 = a^2m^2 - b^2$.Given the line $y = \alpha x + \beta$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,we substitute $m = \alpha$ and $c = \beta$ into the condition.Thus,$\beta^2 = a^2\alpha^2 - b^2$.Replacing $(\alpha, \beta)$ with $(x, y)$ to find the locus,we get $y^2 = a^2x^2 - b^2$.Rearranging the terms,we have $a^2x^2 - y^2 = b^2$,or $\frac{x^2}{(b/a)^2} - \frac{y^2}{b^2} = 1$.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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