Suppose the hypotenuse and its opposite vertex of an isosceles right angled triangle are $3 x+4 y-4=0$ and $(2,2)$ respectively. Then, which of the following is another side of the triangle?

• A

  $x-7 y-12=0$
• B

  $x+7 y+12=0$
• Correct answer
  C

  $7 x+y-16=0$
• D

  $7 x+y+16=0$

Solution

Equation of $B C$ is $3 x+4 y-4=0$
$\therefore$ Slope of $A D=\frac{4}{3}$ Let the slope of one of other sides be $m$. $$ \begin{aligned} & \text { Then, } \tan 45^{\circ}=\left|\frac{m-\frac{4}{3}}{1+\frac{4 m}{3}}\right| \Rightarrow 1=\left|\frac{3 m-4}{3+4 m}\right| \\ & \Rightarrow \quad \frac{3 m-4}{3+4 m}=1 \text { or } \frac{3 m-4}{3+4 m}=-1 \\ & \Rightarrow \quad m=-7 \text { or } m=\frac{1}{7} \end{aligned} $$ and side passes through $(2,2)$. $\therefore$ Equation of side can be $$ 7 x+y-16=0 \text { or } x-7 y+12=0 $$

Hence, the correct answer is option: 'C'.