Question

If the angle of intersection of the circles $x^2+y^2+x+y=0$ and $x^2+y^2+x-y=0$ is $\theta$, then equation of the line passing through $(1,2)$ and making an angle $\theta$ with the y − axis is

• A. $x=1$
• B. $y=2$
• C. $x+y=3$
• D. $x-y=3$

Answer 1

The correct option is B $y=2$

Let $A$ and $B$ be the centers and $r_1$ and $r_2$ be the radii of the two circles. Then,
$A\equiv (-\frac{1}{2},-\frac{1}{2}),B\equiv (-\frac{1}{2},\frac{1}{2})$
$r_1=\frac{1}{\sqrt{2}},r_2=\frac{1}{\sqrt{2}}$
$\cos \theta =\frac{r_1^2+r_2^2-A{B}^2}{2r_1{r}_2}$
$=\frac{\frac{1}{2}+\frac{1}{2}-1}{2\times \frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}}=0$
or $\theta =\frac{\pi }{2}$
Therefore, the required line is parallel to the $x-$ axis and since it passes through $(1,2),$ its equation will be $y=2$.