Find the equation for the ellipse that satisfies the given conditions: $b=3, c=4,$ centre at the origin; foci on the $x$-axis.
(N/A) Given: $b=3, c=4,$ centre at the origin; foci on the $x$-axis.
Since the foci are on the $x$-axis,the major axis is along the $x$-axis.
Therefore,the equation of the ellipse is of the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ where $a$ is the semi-major axis.
We know the relation $a^{2}=b^{2}+c^{2}$.
Substituting the values: $a^{2}=3^{2}+4^{2}=9+16=25$.
Thus,$a^{2}=25$ and $b^{2}=3^{2}=9$.
The equation of the ellipse is $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$.
Since the foci are on the $x$-axis,the major axis is along the $x$-axis.
Therefore,the equation of the ellipse is of the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ where $a$ is the semi-major axis.
We know the relation $a^{2}=b^{2}+c^{2}$.
Substituting the values: $a^{2}=3^{2}+4^{2}=9+16=25$.
Thus,$a^{2}=25$ and $b^{2}=3^{2}=9$.
The equation of the ellipse is $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$.
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