If `P(a sec alpha,b tan alpha)" and "Q(a sec beta, b tan beta)` are two points on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1`. Such that `alpha-beta=2theta`, then PQ touches the hyperbola

Options

A. `(x^(2))/(a^(2)sec^(2)theta)-(y^(2))/(b^(2))=1`
B. `(x^(2))/(a^(2))-(y^(2))/(b^(2)sec^(2)theta)=1`
C. `(x^(2))/(a^(2))-(y^(2))/(b^(2))=cos^(2)theta`
D. none of these

Explanation

The equation of chord `PQ` is
`(x)/(a)cos((alpha-beta)/(2))-(y)/(b)sin((alpha+beta)/(2))=cos((alpha+beta)/(2))`
`implies(x)/(a)costheta-(y)/(b)sin((alpha+beta)/(2))=cos((alpha+beta)/(2))`
`implies(x)/(asectheta)sec((alpha+beta)/(2))-(y)/(b)tan((alpha+beta)/(2))=1`
Clearly, it touches the hyperbola `(x^(2))/(a^(2)sec^(2)theta)-(y^(2))/(b^(2))=1`