Consider the ellipse x^2/9 + y^2/4 =1 and the line x/3 + y/2 = 1.

Question

Consider the ellipse

\(\frac{x^2} {9} + \frac{y^2}{4} =1\) and the line

\(\frac{x}{3}\) + \(\frac{y}{2} = 1\).

(a) Find the points where the line intersects the ellipse?

(b) Shade the smaller region bounded by the ellipse and the line.

(c) Find the area of the shaded region.

Answer 1

(a)

\(\frac{x^2}{9} + \frac{9-6x + x^2}{9}\) = 1 ⇒ 2x² - 6x + 9 = 9

⇒ 2x(x - 3) = 0 ⇒ x = 0,3

When x = 0 ⇒ y = 2

When x = 3 ⇒ y = 2 (\(\frac{3-3}{3}\)) = 0

The intersection points are (0,2) and (3,0).

(b)