If $P, Q, R$ are three points on the conic $\frac{l}{r}=1+e\cos\theta$ and the tangent at $Q$ meets $SP$ and $SR$ in $M$ and $N$ so that $SM=SN=l$, where $S$ is the focus, then prove that the chord $PR$ touches the conic
$\frac{l}{r}=1+2e\cos\theta$
MY ATTEMPT:
Let $\alpha, \beta, \gamma$ be vectorial angle of the points $P, Q, R$ respectively.
Equation of tangent at $Q$ is $\frac{l}{r} = e\cos\theta+\cos(\theta-\beta)$
Equation of chord $PR$ is $\frac{l}{r} = \sec{(\frac{\gamma-\alpha}{2})}\cos{(\theta-\frac{\gamma+\alpha}{2})} + e\cos\theta$
Coordinates of $M$ and $N$ are $(l, \alpha)$ and $(l, \gamma)$ respectively.
I don't know how to reduce the equation of chord $PQ$ to a equation which is identical with the equation of a tangent