Question

The combined equation of the diagonals of the square formed by the pairs of lines
$x y+6 y-4 x-24=0$ and
$x y+6 x-4 y-24=0$ is

• A. $x^{2}-y^{2}+2 x-2 y=0$
• B. $x^{2}-y^{2}+2 x+2 y=0$
• C. $x^{2}-y^{2}+x+y=0$
• D. $x^{2}+y^{2}-2 x y+x-y=0$

Answer 1

Answer: (A)

The given equation of pairs of lines are
$x y+6 y-4 x-24=0$
$\Rightarrow (x+6)(y-4)=0$
$\Rightarrow x+6=0$ and $y-4=0\,\,\,...(i)$
and $x y+6 x-4 y-24=0$
$\Rightarrow (x-4)(y+6)=0$
$\Rightarrow x-4=0$ and $y+6=0\,\,\,...(ii)$
So, the vertices of square are $(4,4),(4,-6),(-6,-6)$ and $(-6,4)$.

Now equation of diagonal $A C$ is $y-4=1(x-4)$
$\Rightarrow x-y=0$
and equation of diagonal $B D$ is $y-4=-1(x+6)$
$\Rightarrow x+y+2=0$
So, combined equation of the diagonals is
$(x-y)(x+y+2)=0$
$\Rightarrow x^{2}-y^{2}+2 x-2 y=0 $