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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

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    $y^2=3(x-1)$ or $y^2=-3(x-1)$.? – 2017-02-14
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    we have to find the angle..... answer is $pie - tan^-1(18√3/47)$ – 2017-02-14
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    I mean what's the equation of parabola.? – 2017-02-14
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    oh well in the question they have just given $y^2−3(x−1)$ so I'll probably go with first case possibility of $y^2=3(x−1)$ – 2017-02-14
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    To solve this, you need to know what "common chord" means; you need to know some method to find intersections of this circle and parabola; you need to know some method to find the focus of the parabola; you need to be able to describe the measure of the angle at one vertex of the triangle formed by these three points. Which of these things can you do for yourself, and which do you need help with? If you edit the question to show all the work you can do on your own, you improve the chance that an answer will tell you something you actually needed. – 2017-02-14

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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 $\tan\theta=\frac {\sqrt{3}}{\frac 14}=4\sqrt{3}$ $$\tan 2\theta=\frac {2\cdot 4\sqrt{3}}{1-(4\sqrt{3})^2}=-\frac {8\sqrt{3}}{47}\\ 2\theta=\tan^{-1}\left(-\frac {8\sqrt{3}}{47}\right)$$