If a chord joining `P(a sec theta, a tan theta), Q(a sec alpha, a tan alpha)` on the hyperbola `x^(2)-y^(2) =a^(2)` is the normal at P, then `tan alpha =`

Options

A. `tan theta (4 sec^(2) theta+1)`
B. `tan theta (4 sec^(2) theta -1)`
C. `tan theta (2 sec^(2) theta -1)`
D. `tan theta (1-2 sec^(2) theta)`

Answer

B

Explanation

Slope of chord joining P and Q = slope of normal at P
`:. (tan alpha - tan theta)/(sec alpha - sec theta) =- (tan theta)/(sec theta)`
`:. tan alpha - tan alpha =- k tan theta` and `sec alpha - sec theta = k sec theta (1+k) sec theta = sec alpha` (1)
`:. (1-k) tan theta = tan alpha` (2)
`[(1+k)sec theta]^(2) - [(1-k)tan theta]^(2) = sec^(2) alpha - tan^(2) alpha =1`
`rArr k =- 2 (sec^(2) theta + tan^(2) theta) =- 4 sec^(2) theta +2`
From (2), `tan alpha = tan theta (1+4 sec^(2) theta -2) = tan theta (4 sec^(2) theta -1)`