One diagonal of a square is along the line $8 x-15 y=0$ and one of its vertex is $(1,2)$. Then, the equation of the sides of the square passing through this vertex are

• A

  $23 x+7 y=9,7 x+23 y=53$
• Correct answer
  B

  $23 x-7 y-9=0,7 x+23 y-53=0$
• C

  $23 x-7 y+9=0,7 x+23 y+53=0$
• D

  None of the above

Solution

Slop of $B D$ is $\frac{8}{15}$ and angle made by $B D$ with $A D$ and $D C$ is $45^{\circ}$.

So, let slope of $D C$ be $m$, then $\tan 45^{\circ}= \pm \frac{m-\frac{8}{15}}{1+\frac{8}{15} m}$ $\Rightarrow \quad(15+8 m)= \pm(15 m-8)$ $\Rightarrow \quad m=\frac{23}{7}$ and $-\frac{7}{23}$ Hence, the equation of $D C$ and $A D$ are $y-2=\frac{23}{7}(x-1)$ $\Rightarrow \quad 23 x-7 y-9=0$ and $\quad y-2=-\frac{7}{23}(x-1)$ $\Rightarrow \quad 7 x+23 y-53=0$

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