Question

Which among the following equations represents a pair of straight lines?

• A. $x^2-y^2+x-y+1=0$
• B. $x^2-2x-y+3=0$
• C. $x^2+3xy+2y^2-x-4y-6=0$
• D. $3x^2-4xy-7y^2=0$

Answer 1

The correct option is D

$3x^2-4xy-7y^2=0$

Equations of the form $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ can be the equation of a conic (Circle, Parabola, Ellipse or Hyperbola) or pair of straight lines.

To decide if it represents a conic or pair of straight lines, we will find

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

If $△=0$, the given equation will represent a pair of straight lines. We will find $△$ for each equation and decide.

1) $x^2-y^2+x-y+1=0$

$a=1,b=-1,h=0,g=\frac{1}{2},f=\frac{-1}{2},c=1$

$\Rightarrow △=1\times 1\times 1+0-{\left(\frac{-1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2-1\times 0=1$

$△\ne 0\Rightarrow conic$

2) $x^2-2x-y+3=0$

$a=1,b=0,h=0,g=-1,f=\frac{-1}{2},c=3$

$△=0+0-1\times {\left(\frac{-1}{2}\right)}^2-0-0$

$=\frac{-1}{4}$

$△\ne 0\Rightarrow conic$

3) $x^2+3xy+2y^2-x-4y-6=0$

$a=1,b=2,h=\frac{3}{2},g=\frac{-1}{2},f=-2,c=-6$

$△=1\times 2\times \frac{3}{2}+2\times -2\times \frac{-1}{2}\times \frac{3}{2}-1\times (-2{)}^2-2\times {\left(\frac{-1}{2}\right)}^2-(-6)\times {\left(\frac{3}{2}\right)}^2$

$3+3-4-\frac{1}{2}+\frac{27}{2}$

$=9-4+13$

$△\ne 0\Rightarrow conic$

4) $3x^2-4xy-7y^2=0$

$a=3,b=-7,h=-2,g=0,f=c=0$

$△=0\Rightarrow$ pair of straight lines

So, only the last equation represents a pair of straight lines.