$PQ:2x+y+6=0$ is a chord hyperbola $x^2-4y^2=4$, and $R=(\alpha,\beta)$ with $\alpha^2+\beta^2-1\leq0$, such that area of triangle $PQR$ is minimum.

Question

Let $PQ: 2x+y+6=0$ is a chord of the curve $x^2 – 4 y^2 = 4$. Coordinates of the point $R(\alpha,\beta)$ that satisfy $\alpha^2+\beta^2-1 \leq 0$, such that area of triangle $PQR$ is minimum ; are given by :

I have tried finding out the points of contact of the hyperbola but for the given equations the points are lengthy to find.

I have also considered drawing a line segment perpendicular to point P to the circle where there which can give minimum area of the triangle

This is a JEE Problem (High-school Level) Where we get 3 Minutes time to solve. Any easily approachable Answer is much appreciated.

Answer 1

Hint1: All the different triangles have the same base. So the one with the least area has the least height.

Hint2: Find the normal of the circle whose slope is perpendicular to the line

Note that you don't have to find $P,Q$

Graph for reference