Question
The length of the common chord of the parabola $$2{y^2} = 3\left( {x + 1} \right)$$ and the circle $${x^2} + {y^2} + 2x = 0$$ is :
A.
$$\sqrt 3 $$
B.
$$2\sqrt 3 $$
C.
$$\frac{{\sqrt 3 }}{2}$$
D.
none of these
Answer :
$$\sqrt 3 $$
Solution :
Solving the equations,
$$\eqalign{
& {x^2} + \frac{{3\left( {x + 1} \right)}}{2} + 2x = 0 \cr
& {\text{or, }}{x^2} + \frac{{7x}}{2} + \frac{3}{2} = 0 \cr
& {\text{or, }}2{x^2} + 7x + 3 = 0 \cr
& {\text{or, }}\left( {2x + 1} \right)\left( {x + 3} \right) = 0 \cr
& {\text{or, }}x = - \frac{1}{2},\, - 3 \cr} $$
But $$x = - 3$$ makes $$y$$ imaginary because $$2{y^2} = 3\left( {x + 1} \right)$$
So, $$x = - \frac{1}{2}\,\,\,\,\,\, \Rightarrow y = \pm \frac{{\sqrt 3 }}{2}$$
$$\therefore $$ the length of the chord $$=$$ the distance between $$\left( { - \frac{1}{2},\,\frac{{\sqrt 3 }}{2}\,} \right)$$ and $$\left( { - \frac{1}{2},\, - \frac{{\sqrt 3 }}{2}\,} \right).$$