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

Find the general equation of the ellipse passing though points A and B, center at-example-1
User Nick Snick
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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

User Francesc VE
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