Question
The equation $$8{x^2} + 8xy + 2{y^2} + 26x + 13y + 15 = 0$$ represents a pair of straight lines. The distance between them is :
A.
$$\frac{7}{{\sqrt 5 }}$$
B.
$$\frac{7}{{2\sqrt 5 }}$$
C.
$$\frac{{\sqrt 7 }}{5}$$
D.
None of these
Answer :
$$\frac{7}{{2\sqrt 5 }}$$
Solution :
The distance between the parallel straight lines given by
$$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$ is $$2\sqrt {\frac{{{g^2} - ac}}{{a\left( {a + b} \right)}}} $$
Here, $$a = 8,\,b = 2,\,c = 15,\,g = 13.$$
So, required distance
$$ = 2\sqrt {\frac{{169 - 120}}{{80}}} = 2 \times \frac{7}{{4\sqrt 5 }} = \frac{7}{{2\sqrt 5 }}$$