Consider the ellipse 9x^2+y^2=36 . Which of the following statements…

The correct answer is Option (1) → (A), (C) and (D) only

Dividing by 36 gives the standard form

    \( \displaystyle \frac{x^{2}}{4}+\frac{y^{2}}{36} = 1 \)

so a = 2 and b = 6. The area enclosed by an ellipse is \(\pi ab\):

    \(A = \pi\times2\times6 = 12\pi\)

By symmetry about both axes, the first-quadrant part is one quarter of it:

    \( \displaystyle \frac{12\pi}{4} = 3\pi \)

A circle \(x^{2}+y^{2} = r^{2}\) encloses the area \(\pi r^{2}\):

    \(x^{2}+y^{2} = 36:\ \ 36\pi\)
    \(x^{2}+y^{2} = 12:\ \ 12\pi\)

So the ellipse encloses \(12\pi\), its first-quadrant part is \(3\pi\), and the two circles enclose \(36\pi\) and \(12\pi\).

Checking the statements

(A) The area of the region enclosed by the ellipse is \(12\pi\) sq. units — Correct. \(\pi ab = \pi\times2\times6 = 12\pi\).

(B) The area of the part of the region in the first quadrant is \(6\pi\) sq. units — Incorrect. It is one quarter of \(12\pi\), which is \(3\pi\).

(C) The area enclosed by the ellipse is less than the area enclosed by the circle \(x^{2}+y^{2}=36\) — Correct. \(12\pi \lt 36\pi\).

(D) The area enclosed by the ellipse is equal to the area enclosed by the circle \(x^{2}+y^{2}=12\) — Correct. That circle encloses \(\pi\times12 = 12\pi\).