Question

The equation of the line of the shortest distance between the parabola $y^2=4x$ and the circle $x^2+y^2-4x-2y+4=0$ is

• A. $x+y=3$
• B. $x-y=3$
• C. $2x+y=5$
• D. none of these

Answer 1

The correct option is A $x+y=3$

Line which is of shortest distance mean to the common normal to those curves.

$C_1:y^2=4x............\left(1\right)$

Normal to it $=>y=mx-2\left(1\right)m-m^3$ ($slopes=m,a=1$)

$=>y=mx=2m-m^3............\left(2\right)$

$C_2:x^2+y^2-4x-2y+4=0...............\left(3\right)$

$C_2:(x-2{)}^2+(y-1{)}^2-1=0$

Each normal to circle passes through its center. If $\left(2\right)$ is normal to circle then it must pass through its center.

$=>1=m\left(2\right)-2m-m^3$

$=>m^3=-1$

So, the line becomes $y=-x+2+1$

$=>y+x=3.$