Question

Tangents to the ellipse $b^2{x}^2+a^2{y}^2=a^2{b}^2$ makes angles ${\theta }_1$ and ${\theta }_2$ with major axis such that $\cot {\theta }_1+\cot {\theta }_2=t$, Then the locus of the point of intersection is

• A. $xy=2t(y^2+b^2)$
• B. $2xy=t(y^2-b^2)$
• C. $4xy=t(y^2-b^2)$
• D. $8xy=t(y^2-b^2)$

Answer 1

The correct option is B $2xy=t(y^2-b^2)$

Let the point of intersection of tangent be $(h,k)$
Then equation of tangent with slope $m$ is given by $y=mx+\sqrt{a^2{m}^2+b^2}$
$(h,k)$ lies on the tangent
then $k=mh+\sqrt{a^2{m}^2+b^2}$
$(a^2-h^2)m^2+2hkm+b^2-k^2=0$
$m_1{+m}_2=-\frac{2hk}{a^2-h^2}$ and $m_1{m}_2=\frac{b^2-k^2}{a^2-h^2}$
given that $\cot {\theta }_1+\cot {\theta }_2=t$
$\Rightarrow \frac{m_1{+m}_2}{m_1{m}_2}=t$
$\Rightarrow \frac{-2hk}{b^2-k^2}=t$
Therefore, $2xy=t(y^2-b^2)$