Question

The distance between the pair of parallel lines $x^2+2xy+y^2-8ax-8ay-9a^2=0$ is

• A. $2\sqrt{5}a$
• B. $\sqrt{10}a$
• C. $10a$
• D. $5\sqrt{2}$

Answer 1

The correct option is D

$5\sqrt{2}$

Explanation for the correct option:

Finding the distance between the pair of parallel lines:

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

The given equation is of the form $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

By comparing both the equations we get,

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

Distance between the parallel lines is given by,

$\begin{array}{rcl}d& =& 2\sqrt{\frac{\left(g^2-ac\right)}{a\left(a+b\right)}}\\ & =& 2\sqrt{\frac{\left({\left(-4a\right)}^2-\left(-9a^2\right)\right)}{1\left(1+1\right)}}\\ & =& 2\sqrt{\frac{16a^2+9a^2}{2}}\\ & =& 2\sqrt{\frac{25a^2}{2}}\\ & =& 2\left(\frac{5a}{\sqrt{2}}\right)\\ & =& 5\sqrt{2}a\\ & =& 5\sqrt{2}\left[\because a=1\right]\end{array}$

The distance between the pair of parallel lines $x^2+2xy+y^2-8ax-8ay-9a^2=0$ is $5\sqrt{2}$

Hence, option (D) is the correct answer.