Find angle subtended by the common chord of circle and parabola at the focus of parabola

Question

Find angle subtended by the common chord of circle $x^2 + y^2 - 2x - 3 = 0$ and parabola $y^2 - 3(x-1)$ at the focus of parabola

Answer 1

Circle: $$x^2+y^2-2x-3=0\\ x^2-2x+1+y^2-4=0\\ (x-1)^2+y^2=4$$ i.e. Circle has centre at $(1,0)$ and radius $2$.

Parabola: $$y^2=3(k-1)\\ y^2=4\left(\frac 34\right) (k-1)$$ i.e. Parabola has focus at $\left(1+\frac 34, 0\right)=\left(\frac 74,0\right)$

As the centre of the circle and the focus of the parabola (with opening in the right direction) lie on the $x$-axis, the common chord must be perpendicular to the $x$-axis. Let the common chord be $x=k$. Substituting this in the formula for the circle and the parabola respectively and equating $y^2$ gives $$4-(k-1)^2=3(k-1)\\ (k-1)^2+3(k-1)-4=0\\ (\overline{k-1}+4)(\overline{k-1}-1)=0\\ k-1=-4, 1\\ k=-3 (\text{not possible as } k>1), 2$$ Hence common chord is $x=2$. Intersection points are $(2,\pm \sqrt3)$.

Angle subtended by common chord at parabola focus is $2\theta$ where $\tan\theta=\frac {\sqrt{3}}{\frac 14}=4\sqrt{3}$ $$\tan 2\theta=\frac {2\cdot 4\sqrt{3}}{1-(4\sqrt{3})^2}=-\frac {8\sqrt{3}}{47}\\ 2\theta=\tan^{-1}\left(-\frac {8\sqrt{3}}{47}\right)$$