Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Explain solution RD Sharma Class 12 Chapter 20 Areas of Bounded Regions exercise 20.3 question 37 Maths.

Explain solution RD Sharma Class 12 Chapter 20 Areas of Bounded Regions exercise 20.3 question 37 Maths.

Answers (1)

Answer:

\frac{32\pi}{3}-\frac{4}{\sqrt{3}} \; \text{sq.units} 

Hints:

Given: Equation of circle x^2+y^2=32  exterior to parabola y^2=6x

Solution: First finding intersection point by solving the equation of two curves

\begin{aligned} &x^{2}+y^{2}=16 \ldots \text { (i) } \\ &\text { and } y^{2}=6 x \ldots \text { (ii) } \\ &\Rightarrow x^{2}+6 x=16 \\ &\Rightarrow x^{2}+6 x-16=0 \end{aligned}

\begin{aligned} &\Rightarrow x^{2}+8 x-2 x-16=0 \\ &\Rightarrow x(x+8)-2(x+8)=0 \\ &\Rightarrow(x+8)(x-2)=0 \\ & x=-8 \end{aligned}

(not possible y^2 cannot be -ve )

Or  x=2    (only allowed value)

y=\pm 2\sqrt{3}

\begin{aligned} &A_{1}=\int_{0}^{2} \sqrt{6 x} d x=\left[\frac{\sqrt{6} x^{\frac{3}{2}}}{\frac{3}{2}}\right]_{0}^{2} \\ &=\frac{2 \sqrt{6}}{3} \times 2^{\frac{3}{2}}=\frac{2 \sqrt{3} \times \sqrt{2}}{3} \times 2 \sqrt{2} \\ &=\frac{8}{\sqrt{3}} \text { sq.units } \end{aligned}

\begin{aligned} &A_{2}=\int_{2}^{4}\left(\sqrt{16-x^{2}}\right) d x=\left[\frac{x}{2} \sqrt{16-x^{2}}+\frac{16}{2} \sin ^{-1} \frac{x}{4}\right]_{2}^{4} \\ &=0+8 \sin ^{-1}-8 \sin ^{-1} \frac{1}{2}-\sqrt{16-4} \\ &=8 \times \frac{\pi}{2}-8 \times \frac{\pi}{6}-\sqrt{12} \\ &=4 \pi-\frac{4 \pi}{3}-\sqrt{12} \\ &=\frac{8 \pi}{3}-\sqrt{12} \text { sq.units } \end{aligned}

\begin{aligned} &\text { Area }=2\left(A_{1}+A_{2}\right) \\ &=2\left(\frac{8}{\sqrt{3}}+\frac{8 \pi}{3}-\sqrt{12}\right) \\ &\text { Required area }=\pi \times 16-\frac{16 x}{3}+4 \sqrt{3}-\frac{16}{\sqrt{3}} \\ &=\frac{32 \pi}{3}-\frac{4}{\sqrt{3}} \end{aligned}\frac{32\pi}{3}-\frac{4}{\sqrt{3}} \; \text{sq.units}

Therefore, the area of the circle is \frac{32\pi}{3}-\frac{4}{\sqrt{3}} \; \text{sq.units} 

Posted by

infoexpert24

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Date Sheet 2025 (Out) Live: Class 10, 12 board exam time table at cbse.gov.in; syllabus, sample papers
anam-khan

CBSE Class 12 board exam date sheet 2025 out; exams from February 15 to April 4
anam-khan

When will CBSE Board exam 2025 date sheet be released for Class 10, 12?

Latest Question