Question

lf the equation of the pair of straight lines passing through the point $(1,1)$ , one making an angle ` $\theta$' with the postive direction of x-axis and the other making the same angle with the positive direction of y-axis is $x^2-(a+2)xy+y^2+a(x+y-1)=0$, $a\ne -2$, then the value of $\sin 2\theta$ is

• A. $a-2$
• B. $a+2$
• C. $\frac{2}{a+2}$
• D. $\frac{2}{a}$

Answer 1

Answer: (C)

$\frac{2}{a+2}$

Equations of the given lines are
$y-1=\tan \theta (x-1)$ and $y-1=\cot \theta (x-1)$
Their combined equation is
$(y-1-\tan \theta (x-1))(y-1-\cot \theta (x-1))=0\Rightarrow x^2-(\tan \theta +\cot \theta )xy+y^2+(\tan \theta +\cot \theta -2)(x+y-1)=0$
Comparing this with given equation we get
$\tan \theta +\cot \theta =a+2\Rightarrow \frac{1}{\sin \theta \cos \theta }=a+2\Rightarrow \sin 2\theta =\frac{2}{a+2}$