Question

If the tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ make angles $\alpha$ and $\beta$ with the major axis such that $\tan \alpha +\tan \beta =\lambda$, then the locus of their point of intersection is

• A. $x^2+y^2=a^2$
• B. $x^2+y^2=b^2$
• C. $x^2-a^2=2\lambda xy$
• D. $\lambda (x^2-a^2)=2xy$

Answer 1

The correct option is D $\lambda (x^2-a^2)=2xy$

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 $k=mh+\sqrt{a^2{m}^2+b^2}$
or $(a^2-h^2)m^2+2hkm+b^2-k^2=0$
Now given $\tan \alpha +\tan \beta =\lambda$
$\Rightarrow m_1+m_2=\lambda$
$\Rightarrow \frac{-2hk}{a^2-h^2}=\lambda$
$\Rightarrow$ Locus is $\lambda (x^2-a^2)=2xy$