Examples and Solutions
Example 1
Find the points of intersection of the circle and the ellipse given by their equations as follows:
\[ x^2 + y^2 = 4 \] \[ \frac{x^2}{4} + \left(y - 1\right)^2 = 9 \]View Solution
We need to solve the system of equations given above.
We first multiply all terms of the second equation by \(-4\) and simplify to obtain:
\[ x^2 + y^2 = 4 \] \[ -x^2 - \left(\frac{4}{9}\right) \left(y - 1\right)^2 = -4 \]We now add side by side the two equations above to obtain a linear equation:
\[ y^2 - \left(\frac{4}{9}\right) \left(y - 1\right)^2 = 0 \]Which may be written as:
\[ 5y^2 + 8y - 4 = 0 \]Solve the quadratic equation for \(y\) to obtain two solutions:
\[ y = -2 \quad \text{and} \quad y = \frac{2}{5} \]We now substitute the value \(y = -2\) into the equation \(x^2 + y^2 = 4\) and solve it for \(x\):
\[ x^2 + (-2)^2 = 4 \implies x = 0 \]We now substitute the value \(y = \frac{2}{5}\) into the equation \(x^2 + y^2 = 4\) and solve it for \(x\):
\[ x^2 + \left(\frac{2}{5}\right)^2 = 4 \] \[ x = \frac{4\sqrt{6}}{5} \approx 1.96 \quad \text{and} \quad x = -\frac{4\sqrt{6}}{5} \approx -1.96 \]The points of intersection of the ellipse and the circle are:
\[ (-2, 0), \quad \left(-\frac{4\sqrt{6}}{5}, \frac{2}{5}\right), \quad \left(\frac{4\sqrt{6}}{5}, \frac{2}{5}\right) \]Shown below is the graph of a circle and an ellipse and their points of intersection:
