Question

If the pair of straight lines ${x^2} - 2pxy - {y^2} = 0$and ${x^2} - 2qxy - {y^2} = 0$, be such that each pair bisects the angle between the other pair, then
A) $pq + 1 = 0$
B) $pq - 1 = 0$
C) $p + q = 0$
D) $p - q = 0$

Answer 1

Hint: You have given two pairs of straight lines and they are asked to find what is the relation between $p$ and $q$ . So, in order to find the solution you have to remember the formula for angle bisector. After that you have to compare the given equations with the formula of angle bisectors. Then derive the equation of angle bisector. Then you have to compare the final two equations to get the right answer.

Complete step by step answer:
First of all let’s see the formula of angle bisector for $a{x^2} + 2hxy + b{y^2} = 0$ is ,
\[ \Rightarrow \dfrac{{({x^2} - {y^2})}}{{(a - b)}} = \dfrac{{xy}}{h}\]
Now, let’s see given equations
First equation of straight line,
$ \Rightarrow {x^2} - 2pxy - {y^2} = 0$
Second given equation of straight line,
$ \Rightarrow {x^2} - 2qxy - {y^2} = 0$
Now, apply formula of angle bisector for ${x^2} - 2pxy - {y^2} = 0$
So we will get,
$a = 1$ , $b = - 1$ , $h = - p$
Equation of angle bisector is,
\[ \Rightarrow \dfrac{{({x^2} - {y^2})}}{{(1 - ( - 1)}} = \dfrac{{xy}}{{ - p}}\]
\[ \Rightarrow ({x^2} - {y^2}) + 2\dfrac{{xy}}{p} = 0\]
Say above expression as equation number $i$
Now, see second equation,
$ \Rightarrow {x^2} - 2qxy - {y^2} = 0$
Now comparing equation number $i$ with second equation we will get results as bellow,
$ \Rightarrow \dfrac{1}{p} = - q$
Do some simple mathematics and we will get,
$ \Rightarrow 1 = - pq$
$ \Rightarrow pq + 1 = 0$
Therefore, option (A) $pq + 1 = 0$ is correct.

Note:
Whenever we face such types of questions the key concept we have to remember the generalized equation of pair of straight lines and the angle between the pair of straight lines and angle bisector which is all stated above, the first compare the given equation with the standard equation then find angle bisector after that compare equations and again and do some simplification.