The distance between the two lines represented by $8 x^2-24 x y+18 y^2-6 x+9 y-5=0$ is
The distance between the two lines represented by $8 x^2-24 x y+18 y^2-6 x+9 y-5=0$ is
0
$\frac{3}{4 \sqrt{13}}$
$\frac{6}{\sqrt{13}}$
$\frac{7}{2 \sqrt{13}}$
Solution
The given equation of family of lines
$8 x^2-24 x y+18 y^2-6 x+9 y-5=0$ $\ldots$ (i)
Comparing with
$a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$
Then, $\left\{\begin{array}{c}a=8, h=-12, b=18 \\ g=-3, f=9 / 2, c=-5\end{array}\right.$
We know that, the distance between two lines represented by
$8 x^2-24 x y+18 y^2-6 x+9 y-5=0 \text { is }$
$2 \sqrt{\left[\frac{\left(g^2-a c\right)}{a(a+b)}\right]}=2 \sqrt{\left\{\frac{9+40}{8(8+18)}\right\}}$
$=2 \sqrt{\frac{49}{208}}=\frac{7}{\sqrt{52}}$
$=\frac{7}{2 \sqrt{13}}$