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The tangent at P on the hyperbola (x^(2)...

The tangent at P on the hyperbola `(x^(2))/(a^(2)) -(y^(2))/(b^(2))=1` meets one of the asymptote in Q. Then the locus of the mid-point of PQ is

A

`3((x^(2))/(a^(2))-(y^(2))/(b^(2)))=4`

B

`(x^(2))/(a^(2))-(y^(2))/(b^(2)) =2`

C

`(x^(2))/(a^(2)) -(y^(2))/(b^(2)) =(1)/(2)`

D

`4((x^(2))/(a^(2))-(y^(2))/(b^(2)))=3`

Text Solution

Verified by Experts

The correct Answer is:
D
Tangent at point `P(a sec theta, b tan theta)` to the hyperbola `(x^(2))/(a^(2)) -(y^(2))/(b^(2)) =1`
`(x sec theta)/(a) - y (tan theta)/(b) =1` (1)
Equation of one of asymptote `rArr (x)/(a) +(y)/(b) =0` (2)
`rArr` Coordinates of Q on the asymptote are `(a(sec theta -tan theta), (-b(sec theta -tan theta))`
Let mid-point of PQ is `M(h,k)`
`rArr h = (a sec theta+(a sec theta -a tan theta))/(2)`
`rArr (h)/(a) = sec theta - (tan theta)/(2)` (3)
Similarly, `(k)/(b) = tan theta - (sec theta)/(2)` (4)
`(3)+(4) rArr (h)/(a) + (k)/(b) = (sec theta + tan theta)/(2)` (5)
`(3)-(4) rArr (h)/(a) - (k)/(b) = (3)/(2) [sec theta - tan theta]` (6)
Now `(5) xx (6) rArr (h^(2))/(a^(2)) - (k^(2))/(b^(2)) = (3)/(4)`
or, `4((x^(2))/(a^(2))-(y^(2))/(b^(2))) =3`
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