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If the chords of contact of tangents from two points `(-4,2)` and `(2,1)` to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` are at right angle, then find then find the eccentricity of the hyperbola.

Text Solution

Verified by Experts

The correct Answer is:
`sqrt((3)/(2))`
Equation of chord of contact with respect to point `P(-4,2)` is
`-(4x)/(a^(2))-(2y)/(b^(2))=1` and that with respect to point (2, 1) is
`(2x)/(a^(2))-(y)/(b^(2))=1`.
According to given condition,
`(((4)/(a^(2)))/((-2)/(b^(2))))xx(((-2)/(a^(2)))/((-1)/(b^(2))))=-1`
`rArr" "(b^(4))/(a^(4))=(1)/(4)`
`rArr" "(b^(2))/(a^(2))=(1)/(2)`
`therefore" "e=sqrt(1+(b^(2))/(a^(2)))=sqrt(1+(1)/(2))=sqrt((3)/(2))`
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