Question

Intersecting Ellipses

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection.

$\left\{\begin{array}{l}4x^2+y^2=4\\ 4x^2+9y^2=36\end{array}\right.$

Short answer

The pair of points of intersection as $(0,2)\&(0,-2)$

The graph is

Step-by-step solution

Step 1. Given information

Equations of ellipses are given as-

$\begin{array}{l}4x^2+y^2=4\\ 4x^2+9y^2=36\end{array}$

Step 2. Concept used

The solutions of the ellipses are the locations at which the equations of ellipses intersect.

In this problem, we have to find the intersection points of the pair of ellipses, sketch the graphs of each pair of equations and label the points of intersection.

Step 3. Calculation

$\begin{array}{l}4x^2+y^2=\mathrm{4....}\left(1\right)\\ 4x^2+9y^2=\mathrm{36....}\left(2\right)\end{array}$

Subtracting equation 1 from equation 2

$\begin{array}{l}4x^2+y^2=\mathrm{4....}\left(1\right)\\ 4x^2+9y^2=\mathrm{36....}\left(2\right)\end{array}$

Putting $y=2$in the equation 1 we get $x=0$

Putting $y=-2$in the equation 1 we get $x=0$

Therefore the solutions of the given equation of ellipses are $(0,2)$ and $(0,-2)$.