Find the equation for the ellipse that satisfies the given conditi | Filo

Class 11

Math

3D Geometry

Conic Sections

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Find the equation for the ellipse that satisfies the given conditions: Foci

Solution: It is given that, foci $$\displaystyle \left ( \pm 3, 0 \right ), a = 4$$
Since the foci are on the $$x$$-axis the major axis is along the$$x$$-axis.
Therefore, the equation of the ellipse will be of the form $$\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1$$ where $$a$$ is the semi-major axis
Accordingly $$ae = 3$$ and $$a = 4$$
We know that $$\displaystyle a^{2}= b^{2}+ a^{2}e^2$$
$$\displaystyle \therefore 4^{2}=b^{2}+3^{2}$$
$$\displaystyle \Rightarrow 16 = b^{2}+9$$
$$\displaystyle \Rightarrow$$ $$\displaystyle b^{2}=16 - 9 = 7$$
Thus, the equation of the ellipse is $$\displaystyle \frac{x^{2}}{16}+\frac{y^{2}}{7}= 1$$
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