The locus of a point P(α,β) moving under the condition that the line y = αx + β is a tangent to the hyperbola x2a2−y2b2=1.
a hyperbola
Here we are dealing with a locus of moving point α,β.
Given that α,β satisfies the condition that y=αx+β is a tangent to hyperbola x2a2−y2b2=1
We known the general form of tangent of slope 'm' which is y=±√a2m2−b2
Since y=αx+β ia a tangent, we can compare the coefficients of the the 2 equations and equate them if
coefficient of one is the same.Since coefficient of y is 1 in both equations,
α=m
β=√a2m2−b2
i.e.,β=√a2α2−b2
i.e.,β2=a2α2−b2
a2α2−b2=b2
since αβ is the locus points. this represents the curve,
i.e.,x2(ab)2−y2b2=1 which is a hyperbola
Hence option (a) is correct.