Question

If the pair of lines $a{x}^2+2(a+b)xy+b{y}^2=0$ 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:

• A. $3a^2+2ab+3b^2=0$
• B. $3a^2+10ab+3b^2=0$
• C. $3a^2-2ab+3b^2=0$
• D. $3a^2-10ab+3b^2=0$

Answer 1

Answer: (A)

Equation of pair of lines is $a{x}^2+2(a+b)xy+b{y}^2=0$
Since $4\theta =\pi \Rightarrow \theta =\frac{\pi }{4}$
Angle between lines is given by
$tan\theta =\frac{2\sqrt{H^2-AB}}{|A+B|}$
$\Rightarrow 1=\frac{2\sqrt{{\left(a+b\right)}^2-ab}}{|a+b|}$
$\Rightarrow a^2+b^2+2ab=4\left(a^2+b^2+ab\right)$
$\Rightarrow 3a^2+3b^2+2ab=0$