Question

If the image of the point $(4,-6)$ by a line is the point $(2,2)$, then the equation of the mirror is

• A. $4x+3y-5=0$
• B. $x-4y=11$
• C. $x+4y-5=0$
• D. $-x+y+11=0$

Answer 1

Answer: (B)

$x-4y=11$

From given points, required line and line joining points are perpendicular to each other,

$\therefore$ slope of required line is $m=-\frac{2-4}{2-(-6)}=\frac{1}{4}$

The midpoint of the given points will be on our line.

$\Rightarrow$$X=\left(\frac{x_1+x_2}{2}\right)andY=\left(\frac{y_1+y_2}{2}\right)$

$\Rightarrow$$X=\left(\frac{4+2}{2}\right)=3andY=\left(\frac{-6+2}{2}\right)=-2$

$\therefore$ point is $(3,-2)\equiv (x_1,y_1)$.

$\therefore$ Equation of line is given by,

$\Rightarrow$$y-y_1=m(x-x_1)$

Then equation of required line is $\frac{y+2}{x-3}=\frac{1}{4}⟹x-4y=11$