Question

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

$⟹S^2=t+t^2+9-6t$

$⟹S^2=t^2-5t+9$

differentiating w.r.t $t$

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

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

$⟹S^2=-\frac{25}{4}+9-9=\frac{11}{4}$

$⟹A=\frac{\sqrt{11}}{2}=\frac{\sqrt{p}}{q}$

$⟹p-q=9$

Hence the value of $p-q$ is $9$.