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Two tangents to the hyperbola (x^(2))/(2...

Two tangents to the hyperbola `(x^(2))/(25) -(y^(2))/(9) =1`, having slopes 2 and m where `(m ne 2)` cuts the axes at four concyclic points then the slope m is/are

A

`-(1)/(2)`

B

`-2`

C

`(1)/(2)`

D

2

Text Solution

Verified by Experts

The correct Answer is:
C
Tangents having slope `'2', y = 2x +- sqrt(100-9) = 2x +- sqrt(91)`
It meets the axis at `A -= (+-(sqrt(91))/(2),0),B (0,+-sqrt(91))`
Tangents having slope `'m', y = mx+- sqrt(25 m^(2)-9)`
It meets the axis at `D(+-(sqrt(25m^(2)-9))/(m),0), C -= (0,+-sqrt(25m^(2)-9))`
ABCD is cyclic quadrilateral
`rArr OA xx OD xx OB xx OC`
`rArr (sqrt(91)(sqrt(25m^(2)-9)))/(2m) =sqrt(91) (sqrt(25m^(2)-9))`
`rArr 2m = 1 rArr m = (1)/(2)`
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