Question

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

• A. $9$
• B. $6$
• C. $4$
• D. $11$

Answer 1

The correct option is A $9$

General point of parabola would be $(\alpha ,{\alpha }^2)$

distance, $S^2={\alpha }^2+(2^2-3{)}^2$

let $2^2=t$

$\Rightarrow S^2=t+t^2+9-6t$

$=t^2-5t+9$

differentiating w.r.t $t$

$2S\frac{ds}{dt}=2t-5=0$

at $t=\frac{5}{2}$

$\Rightarrow$ Minima at $t=\frac{5}{2}$

$\Rightarrow S^2=\frac{25}{4}-\frac{25}{4}+9=9-\frac{25}{4}$

$\Rightarrow S^2=\frac{11}{4}\Rightarrow S=\frac{\sqrt{11}}{2}=\frac{\sqrt{p}}{q}$

$\Rightarrow p-q=9\Rightarrow \left(A\right)$