# 20. Find the equation for the ellipse that satisfies the given conditions:     Major axis on the x-axis and passes through the points (4,3) and (6,2).

Given, in an ellipse

Major axis on the x-axis and passes through the points (4,3) and (6,2).

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

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

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

Now since the ellipse passes through the point,(4,3)

$\frac{4^2}{a^2}+\frac{3^2}{b^2}=1$

${16b^2+9a^2}={a^2b^2}$

since the ellipse also passes through the point (6,2).

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

$4a^2+36b^2=a^2b^2$

On solving this two equation we get

$a^2=52$ and $b^2=13$

Thus, The equation of the ellipse will be

$\frac{x^2}{52}+\frac{y^2}{13}=1$

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