If the tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$  $\left(a>b\right)$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

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  $\tan \alpha +\tan \beta =\lambda$. Then,  $m_1+m_2=\lambda$  
Or  $\frac{-2hk}{a^2-h^2}=\lambda$
Therefore, the locus is $\lambda (x^2-a^2)=2xy$