Straight lines $x-y=7$ and $x+4 y=2$ intersect at $B$. Points $A$ and $C$ are so chosen on these two lines such that $A B=A C$. The equation of line $A C$ passing through (2,-7) is

• Correct answer
  A

  $x-y-9=0$
• Correct answer
  B

  $23 x+7 y+3=0$
• C

  $2 x-y-11=0$
• D

  $7 x-6 y-56=0$

Solution

Given equations of lines are $x-y=7$ ....(i) and $x+4y=2$ ...(ii) By solving Eqs.(i) and (ii), we get the point $B(6,-1)$ Let the slope of $A C$ be $m$ then, $A B=A C$ $\therefore \quad\left|\frac{m+\frac{1}{4}}{1-\frac{m}{4}}\right|=\left|\frac{-\frac{1}{4}-1}{1-\frac{1}{4}}\right|$ $\Rightarrow \quad m=\frac{-23}{7}, 1$ When $m=-\frac{23}{7},$ then equation of line $y+7=\frac{-23}{7}(x-2)$ $\Rightarrow \quad 7 y+49=-23 x+46$ $\Rightarrow \quad 23 x+7 y+3=0$ When $m=1$, then equation of line $y+7=(x-2)$ $\Rightarrow \quad x-y-9=0$

Hence, the correct answer is options: 'A', 'B'.