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10-2. Parabola, Ellipse, Hyperbola
hard
Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2= 16x.$ If one of these tangents touches the two curves at $Q$ and $R$, then $( QR )^2$ is equal to
A
$64$
B
$76$
C
$81$
D
$72$
(JEE MAIN-2023)
Solution
$y = mx +\frac{4}{ m }$
$\frac{\left|\frac{4}{ m }\right|}{\sqrt{1+ m ^2}}=2 \sqrt{2} \therefore m =\pm 1$
$y=\pm x \pm 4$. Point of contact on parabola
Let $m=1,\left(\frac{a}{m^2}, \frac{2 a}{m}\right)$
$R (4,8)$
Point of contact on circle $Q (-2,2)$
$\therefore( QR )^2=36+36=72$
Standard 11
Mathematics
