Question

The point on the curve $y^2=4x$ which is nearest to the point (2, 1) is

• A. $(4,4)$
• B. $(1,2)$
• C. $(9,6)$
• D. $(4,6)$

Answer 1

The correct option is B $(1,2)$

Let $P(x,y)$ be a point on the parabola $y^2=4x$

Distance between $P(x,y)$ and $(2,1)$ is

$S^2=(x-2{)}^2+(y-1{)}^2$

$S^2=(\frac{y^2}{4}-2{)}^2+(y-1{)}^2$

$2S\frac{dS}{dx}=\frac{y^3}{4}-2$

For maxima or minima,

$\frac{dS}{dx}=0$

$\Rightarrow y^3=8$

$\Rightarrow y=2$

$f^{\prime \prime }\left(y\right)>0$ at $y=2$

$\Rightarrow x=1$

Hence, the nearest point is $(1,2).$