# Find the equation for the ellipse that satisfies the given conditions:    Q19.    Centre at (0,0), major axis on the y-axis and passes through the points (3, 2) and (1,6).

Given,in an ellipse

Centre at (0,0), major axis on the y-axis and passes through the points (3, 2) and (1,6).

Since, The major axis of this ellipse is on the Y-axis, the equation of the ellipse will be of the form:

$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

Where $a$ and $b$ are  the length of the semimajor axis and semiminor axis respectively.

Now since the ellipse passes through points,(3, 2)

$\frac{3^2}{b^2}+\frac{2^2}{a^2}=1$

${9a^2+4b^2}={a^2b^2}$

since the ellipse also  passes through points,(1, 6).

$\frac{1^2}{b^2}+\frac{6^2}{a^2}=1$

$a^2+36b^2=a^2b^2$

On solving these two equation we get

$a^2=40$ and $b^2=10$

Thus, The equation of the ellipse will be

$\frac{x^2}{10}+\frac{y^2}{40}=1$.

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