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If the tangents to the ellipse (x^2)/(a^...

If the tangents to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` make angles `alphaa n dbeta` with the major axis such that `tanalpha+tanbeta=gamma,` then the locus of their point of intersection is `x^2+y^2=a^2` (b) `x^2+y^2=b^2` `x^2-a^2=2lambdax y` (d) `lambda(x^2-a^2)=2x y`

A

`x^(2)+y^(2)=a^(2)`

B

`x^(2)+y^(2)=b^(2)`

C

`x^(2)-y^(2)=2lambdaxy`

D

`lambda(x^(2)-b^(2))=2xy`

Text Solution

Verified by Experts

Tangent to the ellipse having slope m is
`y=mx+sqrt(a^(2)m^(2)+b^(2))`
If it passes through the point P(h,k) , then
`k=mh+sqrt(a^(2)m^(2)+n^(2))`
or `(a^(2)-h^(2))m^(2)+2hkm+b^(2)-k^(2)=0`
Now, given, `tan alpha+tan beta=lambda` Then,
`m_(1)+m_(2)=lambda`
or `(-2hk)/(a^(2)-h^(2)=lambda)`
Therfore, the locus is `lambda(x^(2)-a^(2))=2xy`
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