Question

The minimum distance of a point on the curve $y=x^2-4$ from the origin is

• A. $\sqrt{\frac{19}{2}}$
• B. $\frac{\sqrt{19}}{2}$
• C. $\sqrt{\frac{15}{2}}$
• D. $\frac{\sqrt{15}}{2}$

Answer 1

The correct option is D $\frac{\sqrt{15}}{2}$

let the point at the minimum distance from O is $P(h,h^2-4)$

therefore $O{P}^2=h^2+(h^2-4{)}^2$

$\frac{d}{dh}(OP{)}^2=2h+4h(h^2-4)=0$

$=>h=\pm \sqrt{\frac{7}{2}}$

$\frac{d^2}{d{h}^2}\left(O{P}^2\right)$ at $\sqrt{\frac{7}{2}}>0$

$OP$ is minimum at $h=\pm \sqrt{\frac{7}{2}}$

$OPmin=\sqrt{\frac{7}{2}+(\frac{7}{2}-4{)}^2}=\frac{\sqrt{15}}{2}$