Question

Two straight lines are drawn through the point $(0,2)$ such that the length of the perpendiculars from the point $(4,4)$ to these lines are each equal to $2$ units. The equation of the line joining the feet of these perpendiculars is

• A. $y+x=5$
• B. $2 y+3 x=8$
• C. $y-2 x=10$
• D. $y+2 x=10$

Answer 1

Answer: (D)

Let equation of line passing through $(0,2)$ is

$y-2=m x$
$y=m x+2$
$ BC =2 $
$\therefore \, 2=\left|\frac{4 m-4+2}{\sqrt{1+m^{2}}}\right| $
$\sqrt{1+m^{2}} =(2 m-1) $
$ \Rightarrow \,1+m^{2} =4 m^{2}-4 m+1 $
$ \Rightarrow m =0, \frac{3}{4}$
$ m=0, y=2 $
$ \therefore \, D(4,2)$
Equation of line passing through $D$ and $C$ is perpendicular to $AB$.
$\therefore $ Required equation of line
$y-2 =\frac{-(4-0)}{4-2}(x-4) $
$y-2 =-2(x-4) $
$y+2 x =10$