NIMCET 2021 — Mathematics PYQ

1. Equation of Tangent

The equation of any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with slope $m$ is given by:

2. Point of Intersection

Let the point of intersection of two such tangents be $P(h,k)$. Since $P$ lies on the tangent:

3. Squaring both sides

4. Form a Quadratic in $m$

Rearrange the terms to form a quadratic equation in terms of the slope $m$:

5. Condition for Perpendicularity

Since the tangents meet at right angles ($90^{\circ }$), the product of their slopes ($m_1$ and $m_2$) must be $-1$:

From the quadratic equation, the product of roots is $\frac{\text{constant term}}{\text{coefficient of }{m}^2}$:

6. Simplify

Final Answer

Replacing $(h,k)$ with $(x,y)$, the locus is:

This circle is known as the Director Circle of the ellipse. The correct option is 2.