Question

Distance between two lines represented by the pair of straight lines $x^2-6xy+9y^2+3x-9y-4=0$ is

• A. $\frac{1}{2}$
• B. $\frac{5}{\sqrt{2}}$
• C. $\sqrt{10}$
• D. $\sqrt{\frac{5}{2}}$

Answer 1

The correct option is D

$\sqrt{\frac{5}{2}}$

Explanation for the correct options:

Option(D):$\sqrt{\frac{5}{2}}$

Given:$x^2-6xy+9y^2+3x-9y-4=0$

Comparing with $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

We get $a=1,b=9,h=-3,g=\frac{3}{2},c=-4$

Here $h^2=ab$

Distance between two lines represented by the pair of straight lines

$$\begin{gathered} =2\sqrt{\frac{g^2-ac}{a(a+b)}} \\ =2\sqrt{\frac{{\displaystyle \frac{9}{4}}+4}{1(1+9)}} \end{gathered}$$

$$\begin{gathered} =\frac{5}{\sqrt{10}} \\ =\sqrt{\frac{5}{2}} \end{gathered}$$

Hence, Option(D) is correct and Option(A),(B) and (C) are incorrect.