Question:

The parabola y2=xy^2=x divides the circle x2+y2=2x^2 +y^2 = 2 into two parts whose areas are in the ratio

Updated On: Oct 8, 2023
  • 9π+2:3π29\pi + 2 : 3\pi -2
  • 9π2:3π+29\pi - 2 : 3\pi +2
  • 7π2:2π37\pi - 2 : 2\pi -3
  • 7π+2:3π+27\pi + 2 : 3\pi +2
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The Correct Option is B

Solution and Explanation

Areaofcircle=π(2)2=2πArea of circle =\pi\left(\sqrt{2}\right)^{2}=2\pi Area of OCADO=2{AreaofOCAO}OCADO = 2 \left\{Area\, of\, OCAO\right\} =2{areaofOCB+areaofBCA}=2\left\{area of OCB + area of BCA\right\} =201ypdx+2=2\int\limits^{1}_{{0}}y_pdx+212ycdx\int\limits^{{{\sqrt{2}}}}_{{1}}y_cdx where yp=xy_{p}=\sqrt{x} and yc=2x2y_{c}=\sqrt{2-x^{2}} \therefore Required Area =201=2\int\limits^{1}_{{0}}xdx+2\sqrt{x} dx+2 12\int\limits^{{{\sqrt{2}}}}_{{1}}2x2dx\sqrt{2-x^{2}}dx =[23.10]+2[x2x22+sin1x2]12=\left[\frac{2}{3}.1-0\right]+2\left[\frac{x\sqrt{2-x^{2}}}{2}+sin^{-1} \frac{x}{\sqrt{2}}\right]^{^{\sqrt{2}}}_{_{_1}} =43+2{π2π412}=43+2{π412}=3π+26=\frac{4}{3}+2\left\{\frac{\pi}{2}-\frac{\pi }{4}-\frac{1}{2}\right\}=\frac{4}{3}+2\left\{\frac{\pi}{4}-\frac{1}{2}\right\}=\frac{3\pi+2}{6} Bigger area =2π3π+26=9π26=2\pi-\frac{3\pi+2}{6}=\frac{9\pi-2}{6} \therefore Required Ratio =9π23π+2i.e.,9π2:3π+2= \frac{9\pi-2}{3\pi+2} i.e., 9\pi-2 : 3\pi+2
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Concepts Used:

Circle

A circle can be geometrically defined as a combination of all the points which lie at an equal distance from a fixed point called the centre. The concepts of the circle are very important in building a strong foundation in units likes mensuration and coordinate geometry. We use circle formulas in order to calculate the area, diameter, and circumference of a circle. The length between any point on the circle and its centre is its radius. 

Any line that passes through the centre of the circle and connects two points of the circle is the diameter of the circle. The radius is half the length of the diameter of the circle. The area of the circle describes the amount of space that is covered by the circle and the circumference is the length of the boundary of the circle.

Also Check:

Areas Related to Circles Perimeter and Area of CircleCircles Revision Notes