Question

If the equation of the pair of straight lines passing through the point $(1,1)$, one making an angle $\theta$ with the positive 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

The correct option is C $\frac{2}{a+2}$

The lines will be
$y-1=\tan A(x-1)$
and $y-1=\cot A(x-1)$
Therefore their joint equation will be
$(y-1-\cot A(x-1\left)\right)(y-1-\tan A(x-1\left)\right)=0$
$(y-1{)}^2-(\cot A+\tan A\left)\right(x-1\left)\right(y-1)+(x-1{)}^2=0$
$y^2-2y+1-(\cot A+\tan A)(xy-x-y+1)+(x^2-2x+1)=0$
$x^2+y^2-(\cot A+\tan A)\left(xy\right)+\left(\right(\cot A+\tan A)-2)(x+y-1)=0$
Comparing coefficients we get
$\cot A+\tan A=a+2$
$\frac{1}{\sin A\cos A}=a+2$

$2\sin A\cos A=\frac{2}{a+2}$
$=\sin 2A$
$=\sin 2\theta$