Question

If the distance between the pair of parallel lines $x^2+2xy+y^2-8ax-8ay-9a^2=0$ is $25\sqrt{2}$, then $a$ is

• A. $\pm 4$
• B. $\pm 2$
• C. $\pm 3$
• D. $\pm 5$

Answer 1

The correct option is D $\pm 5$

The equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents the general equation of pair of lines which are parallel to each other, then the distance between them is given by,

$d=\left|2\sqrt{\frac{g^2-ac}{a(a+b)}}\right|$ (or) $d=\left|2\sqrt{\frac{f^2-ac}{b(a+b)}}\right|$

Here, the equation is,

$x^2+2xy+y^2-8ax-8ay-9a^2=0$

i.e., $a=1,b=1,c=-9a^2,h=1,f=-4a,g=-4a$

$\because d=25\sqrt{2}$

$⟹25\sqrt{2}=\left|2\sqrt{\frac{(-4a{)}^2-1(-9a^2)}{1(1+1)}}\right|$

$⟹25\sqrt{2}=\left|2\sqrt{\frac{16a^2+9a^2}{2}}\right|$

$⟹25\sqrt{2}=\sqrt{2}\left(5a\right)$

$\therefore a=\pm 5$ (Ans)