Question

The point on the curve $x^2=2y$ which is nearest to $(0,5)$ is :

• A. $(2\sqrt{2},0)$
• B. $(0,0)$
• C. $(2,2)$
• D. None of these

Answer 1

The correct option is D None of these

Let take (h,k) point on the curve which is nearest to the point (0,5).
$d=\sqrt{h^2+{(k-5)}^2}$
As $h^2=2k$
$d=\sqrt{h^2+{(h^2/2-5)}^2}$
Minimize $d^2$
$2x+2(x^2/2-5)\ast 2x/2=0$
$2x+x^3-10x=0$
$x^3-8x=0$
$x=+2\sqrt{2},-2\sqrt{2}$
$-2\sqrt{2}$ is discarded because it will not give the nearest point. Draw the curve and check $.x=+2\sqrt{2}$,$y=4$