Question

A general equation of second degree $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a pair of straight lines if.

• A. $h^2=2ab$
• B. $h^2>2ab$
• C. $\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$
• D. None of these

Answer 1

The correct option is C

$\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$

Explanation for the correct answer:

Condition to represent a pair of straight lines.

The general second order degree equation is $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$.

To represent the pair of straight lines the condition for the second degree equation is

$\Rightarrow abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

we can write this equation in determinant form with the condition, $∆=0$ as

$\Rightarrow \left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$

Hence, option (C) is the correct answer.