Let ‘C’ be the chord of a circle ‘S’ of radius r which subtends angle $\frac{2 π}{3}$ at the centre of S. If ‘R’ represents the region consisting of all points inside S which are closer to ‘C’ than to the circumference of ‘S’, then which is /are CORRECT ?

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C divides the region R in the ratio 1 : 3

Area of R is $\frac{\sqrt{3}}{2} r^{2}$

C divides the region R in the ratio 1 : 4

Area of R is $\frac{2}{\sqrt{3}} r^{2}$

answer is A, C.

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Detailed Solution

R is enclosed by two confocal parabolas.

Focus of each of them is the centre of the given circle,

whereas vertices are lying on the perpendicular bisector of C at distances $\frac{3 r}{4}$ and $\frac{r}{4}$ from centre.

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