# Find the equation of an ellipse that satisfies the following conditions: Centre at $(0,0)$, major axis on y-axis and passes through $(3,2), \; (1,6)$

Toolbox:
• Given an ellipse as follows:
• http://clay6.com/mpaimg/Toolbar_8.png
• The equation of the ellipse is $\large\frac{x^2}{b^2}$$+\large\frac{y^2}{a^2}$$=1$
• Compare the given equation to the general equation of the ellipse to infer $a$ and $b$.
• $c = \sqrt {a^2 - b^2}$
• Given that the major axis is the y-axis, a is the semi-major axis and b the semi-minor axis.
• We can find out the equations of the ellipse in terms of $a^2$ and $b^2$ and solve for them and substitute back in the standard equation of an ellipse.
Given Centre at $(0,0)$, major axis on y-axis and passes through $(3,2), \; (1,6)$
We therefore can write the equation of the ellipse as $\;\large\frac{3^2}{b^2}$$+\large\frac{2^2}{{a}^2}$$=1$ and $\;\large\frac{1^2}{b^2}$$+\large\frac{6^2}{{a}^2}$$=1$
Solving the two equations, we get $a^2 = 40,\; b^2 = 10$
$\Rightarrow$ The equation of the ellipse is $\large\frac{x^2}{10}$$+\large\frac{y^2}{40}$$=1$