Parabola Ques 11
- From a point $A$ common tangents are drawn to the circle $x^2+y^2=\frac{a^2}{2}$ and parabola $y^2=4 a x$. Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of contact of the parabola.
(1996, 2M)
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Answer:
Correct Answer: 11.$\left(\frac{15 a^2}{4}\right)$
Solution: Equation of any tangent to the parabola, $y^2=4 a x$ is $y=m x+\frac{a}{m}$
This line will touch the circle $x^2+y^2=\frac{a^2}{2}$
If $\left(\frac{a}{m}\right)^2=\frac{a^2}{2}\left(m^2+1\right)$
$\Rightarrow \quad \frac{1}{m^2}=\frac{1}{2}\left(m^2+1\right)$
$\Rightarrow \quad 2=m^4+m^2$
$\Rightarrow \quad m^4+m^2-2=0$
$\Rightarrow \quad\left(m^2-1\right)\left(m^2+2\right)=0$
$\Rightarrow \quad m^2-1=0, m^2=-2$
$\Rightarrow \quad m= \pm 1 \quad\left [m^2=-2\right.$ is not possible]
Therefore, two common tangents are
$ y=x+a \text { and } y=-x-a $
These two intersect at $A(-a, 0)$.
The chord of contact of $A(-a, 0)$ for the circle
$x^2+y^2=a^2 / 2$ is $(-a) x+0 \cdot y=a^2 / 2$
$\Rightarrow \quad x=-a / 2$
and chord of contact of $A(-a, 0)$ for the parabola $y^2=4 a x$ is $0 \cdot y=2 a(x-a) \quad \Rightarrow \quad x=a$
Again, length of $B C=2 B K$
$ \begin{aligned} & =2 \sqrt{O B^2-O K^2} \\ & =2 \sqrt{\frac{\alpha^2}{2}-\frac{a^2}{4}}=2 \sqrt{\frac{a^2}{4}}=a \end{aligned} $
and we know that, $D E$ is the latusrectum of the parabola, so its length is $4 a$.
Thus, area of the quadrilateral $B C D E$
$ \begin{aligned} = & \frac{1}{2}(B C+D E)(K L) \\ = & \frac{1}{2}(a+4 a)\left(\frac{3 a}{2}\right)=\frac{15 a^2}{4} \end{aligned} $
BETA
