For the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` , let `n` be the number of points on the plane through which perpendicular tangents are drawn. If `n=1,t h e ne=sqrt(2)` If `n >1,t h e n0sqrt(2)` None of these

Options

A. If n = 1, then `e=sqrt2`
B. If n gt 1, then `0 lt e lt sqrt2.`
C. If n = 0, then `e gt sqrt2`.
D. none of these

Answer

A, B, C

Explanation

The locus of the point of intersection of perpendicular tangents is director circle `x^(2)+y^(2)=a^(2)-b^(2)`. Now,
`e^(2)=1+(b^(2))/(a^(2))`
If `a^(2) gt b^(2)`, then there are infinite (or more than 1) points on the circle, i.e., `e^(2)lt2 or e ltsqrt2`. ltBrgt If `a^(2) lt b^(2)`, there does not exist any point on the plane, i.e.,
`e^(2) gt2 or e gtsqrt2`.
If `a^(2)=b^(2)`, there is exactly one point (center of the hyperbola),
i.e., `e=sqrt2`.