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If a chord joining P(a sec theta, a tan ...

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 ` is (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)`

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)`

Text Solution

Verified by Experts

The correct Answer is:
B
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)`
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