Find the Points of Intersection of Two Ellipses

We first multiply all terms of the first equation by \( 16 \) and all the terms of the second equation by \( -2 \) and simplify to obtain equivalent equations given by:

\[ x^2 + 4(y + 1)^2 = 16 \] \[ -x^2 - \dfrac{1}{6}(y + 2)^2 = -2 \]

We now add side by side the two equations to eliminate \( x^2 \) and obtain a quadratic equation in terms of \( y \):

\[ 4(y + 1)^2 - \dfrac{1}{6}(y + 2)^2 = 14 \]

Multiply all terms by 6, group like terms, and rewrite the equation in standard quadratic form:

\[ 23y^2 + 44y - 64 = 0 \]

Solve the quadratic equation for \( y \) using the quadratic formula to obtain two solutions:

\[ y \approx 0.97 \quad \text{and} \quad y \approx -2.88 \]

We now substitute the values of \( y \) already obtained into the equation \( x^2 + 4(y + 1)^2 = 16 \) and solve it for \( x \) to obtain the corresponding \( x \) values:

• For \( y \approx 0.97 \): \( x \approx 0.730365 \) and \( x \approx -0.730365 \)
• For \( y \approx -2.88 \): \( x \approx 1.36788 \) and \( x \approx -1.36788 \)

Thus, the 4 points of intersection of the two ellipses are:

\[ (0.73, 0.97), \quad (-0.73, 0.97), \quad (1.37, -2.88), \quad (-1.37, -2.88) \]

The graph of the two ellipses and their intersection points is shown below: