Question

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

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

Answer 1

Answer: (A)

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

Now equation of diagonal $AC$ is $y-4=1(x-4)$
$\Rightarrow x-y=0$
and equation of diagonal $BD$ is $y-4=-1(x+6)$
$\Rightarrow x+y+2=0$
So, combined equation of the diagonals is
$(x-y)(x+y+2)=0$
$\Rightarrow x^2-y^2+2x-2y=0$