# Ex 8.2,1

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 8.2 , 1 Find the area of circle 4 𝑥2+4 𝑦2=9 which is interior to the parabola x2 = 4𝑦 Given equations 4 𝑥2+4 𝑦2=9 𝑥2=4𝑦 We can write (1) as 4 𝑥2+4 𝑦2=9 4 𝑥2+4 𝑦24= 94 𝑥2+ 𝑦2= 94 𝑥2+ 𝑦2= 322 Comparing with (𝑥−ℎ)2+ (𝑦−𝑘)2= 𝑟2 , It is a circle with radius (r) = a & center = (h, k) = (0, 0) Also, 𝑥2=4𝑦 This is equation of a parabola with a vertical axis Step 1: Make the figure Required Area = Area ABCO Finding point of intersection A and C Solving (1) and (2) 4 𝑥2+4 𝑦2=9 …(1) 𝑥2=4𝑦 …(2) Putting value of 𝑥2 from (2) in (1) 4 𝑥2+4 𝑦2=9 4 4𝑦+4 𝑦2=9 16𝑦+4 𝑦2−9=0 4𝑦2+16𝑦−9=0 4𝑦2+18𝑦−2𝑦−9=0 2y 2y+9−1 2𝑦+9=0 (2y −1)(2𝑦+9)=0 Hence, 𝑦= 12 & 𝑦= −92 Putting values of y in (2) Hence the points are A= − 2 , 12 & C= 2 , 12 Step 3: Finding Area Area required = Area ABCO Since ABCO is symmetric in y – axis, Area ABCO = 2 × Area BOC Area BOC = Area BCDO − Area OCD Area BCDO Area BCDO = 0 2𝑦 𝑑𝑥 y → equation of circle 4x2 + 4y2 = 9 4y2 = 9 − 4x2 y2 = 94 − x2 y = ± 94 − x2 Since BCDO is above x − axis, we take y positive ∴ y = 94 − x2 So Area BCDO = 0 2 94 − x2 dx = 0 2 322 − x2 dx = 𝑥2 322 − x2+ 3222 sin−1 𝑥 32 0 2 = 𝑥2 322 − x2+ 98 sin−1 2𝑥3 0 2 = 22 94 − (2)2 + 98 sin−1 2 23 − 02 94 − 02+ 98 sin−1 2×03 = 22 94 − 4+ 98 sin−1 2 23− 98 sin−1(0) = 22× 14+ 98 sin−1 2 23− 98×0 = 24+ 98 sin−1 2 23 Area OCD Area OCD = 0 2𝑦 dx y → equation of parabola x2 = 4y y = 𝑥24 So, Area OCD = 0 2𝑦 dx = 0 2 𝑥24 dx = 14 0 2 𝑥2 dx = 14 𝑥330 2 = 14 233− 033 = 14 2× 2× 23−0 = 14 2 23 = 26 So Area BOC = Area BCDO − Area OCD = 24 + 98 sin−1 2 23 – 26 = 24 − 26+ 98 sin−1 2 23 = 212 + 98 sin−1 2 23 Area required = Area ABCO = 2 × Area BOC = 2 × 212+ 98 sin−1 2 23 = 26+ 94 sin−1 2 23

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.