Consider the circles \(C_{1}: x^{2}+y^{2}=1\) and \(C_{2}: x^{2}+y^{2}=9\). Which of the following statements are correct?
(A) The area of the region between \(C_{1}\) and \(C_{2}\) is \(8\pi\) sq. units
(B) The area of the region enclosed by \(C_{1}\) is one-third of the area of the region enclosed by \(C_{2}\)
(C) The area of the part of the region between \(C_{1}\) and \(C_{2}\) that lies in the first quadrant is \(4\pi\) sq. units
(D) The area of the region between \(C_{1}\) and \(C_{2}\) is 8 times the area of the region enclosed by \(C_{1}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (D) only
The circles have the same centre and radii 1 and 3. A circle of radius r encloses the area \(\pi r^{2}\):
\(C_{1}:\ \ \pi\times1^{2} = \pi\)
\(C_{2}:\ \ \pi\times3^{2} = 9\pi\)
The region between them is the larger disc with the smaller one removed:
\(9\pi-\pi = 8\pi\)
The ring is symmetric about both axes, so its first-quadrant part is one quarter of it:
\( \displaystyle \frac{8\pi}{4} = 2\pi \)
So \(C_{1}\) encloses \(\pi\), \(C_{2}\) encloses \(9\pi\), the ring between them has area \(8\pi\), and its first-quadrant part has area \(2\pi\).
Checking the statements
(A) The area of the region between \(C_{1}\) and \(C_{2}\) is \(8\pi\) sq. units — Correct. \(9\pi-\pi = 8\pi\).
(B) The area of the region enclosed by \(C_{1}\) is one-third of the area of the region enclosed by \(C_{2}\) — Incorrect. \(\pi\) is one-ninth of \(9\pi\).
(C) The area of the part of the region between \(C_{1}\) and \(C_{2}\) that lies in the first quadrant is \(4\pi\) sq. units — Incorrect. It is one quarter of \(8\pi\), which is \(2\pi\).
(D) The area of the region between \(C_{1}\) and \(C_{2}\) is 8 times the area of the region enclosed by \(C_{1}\) — Correct. \(8\pi = 8\times\pi\).