Question

The equation $6x^2+xy-12y^2-13x+6y+6=0$ represents

• A. a pair of straight lines through the origin
• B. a pair of perpendicular straight lines
• C. a pair of parallel straight lines
• D. a pair of straight lines not passing through the origin, neither parallel nor perpendicular

Answer 1

The correct option is D a pair of straight lines not passing through the origin, neither parallel nor perpendicular

For the equation
$6x^2+xy-12y^2-13x+6y+6=0$
$a=6,b=-12,c=6,h=1/2,g=-13/2,f=3$
$\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=\left|\begin{array}{ccc}6& 1/2& -13/2\\ 1/2& -12& 3\\ -13/2& 3& 6\end{array}\right|\Rightarrow \left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$
Hence the given line represents a pair of straight lines.
$(0,0)$ doesn't satisfy the given equation.

Let the angle between two lines be $\theta$ so,
$\tan \theta =|\frac{2\sqrt{h^2-ab}}{a+b}|$

So the lines will be perpendicular when,
$a+b=0$
It will be parallel when,
$h^2=ab$

Hence, the given equation represents pair of straight lines not passing through origin, neither parallel nor perpendicular.