Question

Find the general equation of the ellipse passing though points A and B, center at C.

Answer 1

The general equation of an ellipse is

`((x-h)^2)/(a^2)+((y-k)^2)/(b^2)=1`

Where (h, k) are the coordinates of the center.

For our ellipse, we already have the center (2, 1), then, our equation will be

`((x-2)^2)/(a^2)+((y-1)^2)/(b^2)=1`

Using the points that belong to this ellipse, we can substitute them on this equation to find the coefficients a and b.

First, let's use the point (4, 1). Doing the substitution, we have

`\begin{gathered} ((4-2)^2)/(a^2)+((1-1)^2)/(b^2)=1 \\ ((2)^2)/(a^2)=1 \\ a=2 \end{gathered}`

Using the other point, we can calculate the other coefficient.

`\begin{gathered} ((1-2)^2)/(4)+\frac{(1+2\sqrt[]{3}-1)^2}{b^2}=1 \\ ((-1)^2)/(4)+\frac{(2\sqrt[]{3})^2}{b^2}=1 \\ (1)/(4)+\frac{12^{}}{b^2}=1 \\ \frac{12^{}}{b^2}=(3)/(4) \\ 4\cdot12=3\cdot b^2 \\ b^2=16 \\ b=4 \end{gathered}`

And then, we have our ellipse equation.

`((x-2)^2)/(4)+((y-1)^2)/(16)=1`