Question

$a{x}^2+2hxy+b{y}^2=0$ represents a pair of straight lines through origin & angle between them is given by

$\tan \theta =\frac{2\sqrt{h^2-ab}}{a+b}$. If the lines are perpendicular then $a+b=0$ and the equation of bisectors is given by $\frac{x^2-y^2}{a-b}=\frac{xy}{h}$

The general equation of second degree given by

$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represent a pair of straight lines if $△=0$ or

$\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$ or $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

On the basis of above information answer the following question

The Joint equation of the bisectors of the angle between the lines represented by $a{x}^2+2hxy+b{y}^2=0$ is

• A. a pair of perpendicular lines
• B. a pair of parallel lines
• C. a pair of intersecting lines but not $\perp$er
• D. None of these

Answer 1

Answer: (A)

a pair of perpendicular lines

Equation of bisectors of angle between the pair of straight line $a{x}^2+2hxy+b{y}^2=0$ is given by $\frac{x^2-y^2}{a-b}=\frac{xy}{h}$

$\Rightarrow h{x}^2-(a-b)xy-h{y}^2=0$

Here coefficient of $x^2+$ coefficient of $y^2=h+(-h)=0$

$\therefore$ above equation represent the pair of $\perp$er lines

Hence choice (a) is correct.