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The locus a point P(alpha,beta) moving u...

The locus a point `P(alpha,beta)` moving under the condition that the line `y=alphax+beta` is a tangent to the hyperbola `x^2/a^2-y^2/b^2=1` is (A) a parabola (B) an ellipse (C) a hyperbola (D) a circle

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The correct Answer is:
`(x^(2))/(b^(2)//a^(2))-(y^(2))/(b^(2))=1`
The line `y=alphax+beta` touches the hyperbola
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`
If `beta^(2)=a^(2)alpha^(2)-b^(2).`
Hence, the locus of `(alpha, beta)` is
`y^(2)=a^(2)x^(2)-b^(2)`
`"or "a^(2)x^(2)-y^(2)=b^(2)`
`"or "(x^(2))/(b^(2)//a^(2))-(y^(2))/(b^(2))=1`
which is a hyperbola
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