Question

Bed of two rivers are parabola $y^2=4x$ and straight line $y=x+2.$ These rivers are to be connected by a straight canal. The length of the shortest canal possible is

• A. $\sqrt{2}$
• B. $2\sqrt{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $2$

Answer 1

The correct option is C $\frac{1}{\sqrt{2}}$

For the shortest length of canal, the slope of tangent at the closest point will be same for both curve
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{c_1}={\left(\frac{dy}{dx}\right)}_{c_2}$
$\Rightarrow \frac{2}{y}=1$
$\Rightarrow y=2,$ putting in $y^2=4x$
$\Rightarrow x=1$
So, point $(1,2)$ on parabolic curve is nearest to $y=x+2$
So, shortest length=perpendicular distance $=\frac{|2-1-2|}{\sqrt{2}}=\frac{1}{\sqrt{2}}$