Question

If the angle of intersection of the circles x² +y² +x+y = 0 and x² +y² +x–y = 0 is θ , then equation of the line passing through (1,2) and making an angleθ with the y–axis is

Answer 1

Let C₁ and C₂ be the centres of given circles C₁ 

\(\left(\frac{-1}2,\frac{-1}2\right)\) and \(\left(\frac{-1}2,\frac{1}2\right)\)

Also radius these two circles are r₁= \(\sqrt{\frac{1}4+\frac{1}4}=\sqrt{\frac{1}2}=\frac{1}{\sqrt{2}}\)

\(and \,r_2\sqrt{\frac{1}4+\frac{1}4}=\sqrt{\frac{1}2}\)

\(cos\theta=\frac{r_1^2+r_2^2-d^2}{2r_1r_2}\)

\(= \frac{\frac{1}2+\frac{1}2-1}{2\frac{1}{\sqrt{2}}\times\frac{1}{\sqrt{2}}}\)

= 0

∴ θ = π/2 

∴ Required line is parallel to x–axis and it passes through (1, 2)

∴ Equation of line is y = 2.