Question

In a triangle $ABC,$ the bisectors of angles $B$ and $C$ lie along the lines $x=y$ and $y=0$. If $A$ is $(1,2),$ then the equation of line $BC$ is

• A. $2x+y=1$
• B. $x+3y=5$
• C. $x-2y=3$
• D. $x+3y=1$

Answer 1

The correct option is B $x+3y=5$

Reflections of $A$ in the two angle bisectors will lie on the line $BC$. Let the images be$A_1$ and $A_2$ respectively

Formula for image of a point $(h,k)$ in line $ax+by+c=0$ is : $\frac{x-h}{a}=\frac{y-k}{b}=-2\times \frac{ah+bk+c}{a^2+b^2}$ where $(x,y)$ are co-ordinates of image

Image of $A(1,2)$ in $y-x=0$ is $x=1+2\times \frac{-1+2}{2}=2,y=2-2\times \frac{-1+2}{2}=1$ $⟹A_1=(2,1)$

Image of $A$ in $y=0$ is $(-1,2)$

Equation of line $BC$ is : $y-2=\frac{2-1}{-1-2}\times (x+1)$ $⟹x+3y=5$