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. 9
• B. 6
• C. 4
• D. 11

Answer 1

The correct option is A 9

Any point on Parabola is $(\alpha ,{\alpha }^2)$

Distance $S=D^2=(\alpha {)}^2+({\alpha }^2-3^2)={\alpha }^4-5{\alpha }^2+9$

$\frac{dS}{d\alpha }=4{\alpha }^3-10\alpha =0$ at $\alpha (4{\alpha }^2-10)=0$

$\Rightarrow$ at $\alpha =0$ and ${\alpha }^2=5/2$

and $D^2=\frac{5}{2}+(\frac{5}{2}-3{)}^2=\frac{5}{2}+(\frac{1}{2}{)}^2=\frac{10}{4}+\frac{1}{4}=\frac{11}{4}$

$\Rightarrow D=\frac{\sqrt{11}}{2}\Rightarrow$ p=11 q=2 = p-q = 9 = (A)