Q. If the pair of lines $ax^2 + 2(a + b)xy + by^2 = 0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector, then:
AIEEEAIEEE 2005Conic Sections
Solution:
Equation of pair of lines is $ax^2 + 2(a + b) xy + by^2 = 0$
Since $4\theta=\pi \Rightarrow \theta=\frac{\pi}{4}$
Angle between lines is given by
$tan\,\theta=\frac{2\sqrt{H^{2}-AB}}{\left|A+B\right|}$
$\Rightarrow 1=\frac{2\sqrt{\left(a+b\right)^{2}-ab}}{\left|a+b\right|}$
$\Rightarrow a^{2}+b^{2}+2\,ab=4\left(a^{2}+b^{2}+ab\right)$
$\Rightarrow 3a^{2}+3b^{2}+2\,ab=0$
