Points of Intersection of an Ellipse and a Line

We first solve the equation of the line for \( y \) to obtain:

\[ y = 2x - 2 \]

We now substitute \( y \) with \( 2x - 2 \) in the equation of the ellipse:

\[ \dfrac{x^2}{9} + \dfrac{(2x - 2)^2}{4} = 1 \]

Multiply all terms by 36, group like terms, and rewrite the equation as:

\[ 40x^2 - 72x = 0 \]

Solve the quadratic equation for \( x \) to obtain two solutions:

\[ x = 0 \quad \text{and} \quad x = \dfrac{9}{5} \]

We now substitute the values of \( x \) already obtained into the equation \( y = 2x - 2 \) and find \( y \):

• For \( x = 0 \): \( y = 2(0) - 2 = -2 \)
• For \( x = \dfrac{9}{5} \): \( y = 2\left(\dfrac{9}{5}\right) - 2 = \dfrac{18}{5} - \dfrac{10}{5} = \dfrac{8}{5} \)

There are 2 points of intersection given by:

\[ (0, -2) \quad \text{and} \quad \left(\dfrac{9}{5}, \dfrac{8}{5}\right) \]

The graphs of the ellipse and the line given by their equations above and their points of intersection are shown below: