Question

If the shortest distance of (0,3) from parabola y = $x^2$ is $\frac{\sqrt{p}}{q}$ (where p and q are coprime number), then (p-q) is equal to

• A. 8
• B. 6
• C. 4
• D. 11

Answer 1

The correct option is A 8

General point on parabola $=(\alpha ,{\alpha }^2)$

distance, d of this point from (0,3) is,

d $=\sqrt{{\alpha }^2+({\alpha }^2-3{)}^2}\Rightarrow d^2={\alpha }^4-5{\alpha }^2+9$

$\Rightarrow \frac{d}{d\alpha }\left(d^2\right)=4{\alpha }^3-10\alpha =0at2\alpha (2{\alpha }^2-5)=0\Rightarrow \alpha =0,\pm \sqrt{\frac{5}{2}}$

$\Rightarrow \frac{d^2}{d{\alpha }^2}\left(d^2\right)=12{\alpha }^2-10$ At $\alpha =0,\frac{{\partial }^2}{\partial {\alpha }^2}\left(d^2\right)<0\Rightarrow$ Minima

$\Rightarrow d=\frac{\sqrt{9}}{1}\Rightarrow p-q=8$