Find the locus of a point `P(alpha, beta)` moving under the condition that the line `y=ax+beta` is a tangent to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`.

Options

A. a hyperbola
B. a parabola
C. a circle
D. an ellipse

Answer

A

Explanation

If `y=alpha x+beta` touches the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`. ltbtgt then
`beta^(2)=a^(2)alpha^(2)-b^(2)` [Putting `c=beta` and `m=alpha` in `c^(2)=a^(2)m^(2)-b^(2)`]
Hence, locus of `P(alpha,beta)` is
`y^(2)=a^(2)x^(2)-b^(2)` or , `a^(2)x^(2)-y^(2)=b^(2)` which represents a hyperbola.