Question

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

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

Answer 1

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

Let take $(x_1,y_1)$ be the point on the curve $y=x^2-4$ (1)

Another point is $(0,0)$

Distance $=\sqrt{(x_1-0{)}^2+(y_1-0{)}^2}$ (2)

(Distance)${}^2$$=(x_1{)}^2+(y_1{)}^2$

Using $\left(1\right)$

(Distance)${}^2$ $=y+4+y^2$

Differentiate it to give $2y+1=0$

$y=\frac{-1}{2}$

From $\left(1\right)$ $\frac{-1}{2}+4=x^2$

$x=\pm \frac{\sqrt{7}}{\sqrt{2}}$

Since $x$ and $y$ co -ordinates are given.

MInimum Distance $=\sqrt{}=\frac{\sqrt{1}5}{2}$