If the pair of lines ax^2+2(a+b)xy+by^2=0 lie along diameters of a circle and divide the c…

Question

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:

Accepted Answer

If one sector's area is thrice another's, the angle between the two diameters is $45^\circ$ (since $\theta+3\theta=180^\circ\Rightarrow\theta=45^\circ$).

For $ax^2+2(a+b)xy+by^2=0$, $\tan\theta=\dfrac{2\sqrt{h^2-ab}}{a+b}$ with $h=a+b$:
$$\tan45^\circ=1=\frac{2\sqrt{(a+b)^2-ab}}{a+b}\Rightarrow(a+b)^2=4(a^2+ab+b^2)\Rightarrow3a^2+2ab+3b^2=0.$$

Solutions · 1