Question

Find the vertices and foci of the ellipse.49x^2 = 81 - 81y^2

Answer 1

Given the following equation of an ellipse:

$49x^2=81-81y^2$

The standard form equation of an ellipse is:

$\text{ }\frac{(x-h)^2}{a^2}\text{ + }\frac{(y-k)^2}{b^2}\text{ = }1$

Where (h, k) is the center, a and b are the lengths of the semi-major and the semi-minor axis.

Let's transform it into the standard form:

$49x^2=81-81y^2$$49x^2+81y^2=81$$\frac{49x^2+81y^2=81}{81}$$\frac{49x^2}{81}+\frac{y^2}{1}=1$$\frac{x^2}{\frac{81}{49}}+\frac{y^2}{1}=1$$\frac{(x-0)^2^{}}{\frac{81}{49}}+\frac{(y-0)^2^{}}{1}=1$$\frac{(x-0)^2}{(\frac{9}{7})^2}+\frac{(y-0)^2}{1}=1$

From this equation, we get:

$\begin{gathered} \text{ h = 0} \\ \text{ k = 0} \\ \text{ a}^2\text{ = }\frac{81}{49}\text{ }\rightarrow\text{ a = }\sqrt[]{\frac{81}{49}}\text{ }\rightarrow\text{ a = }\frac{9}{7} \\ \text{ }b^2\text{ = 1 }\rightarrow\text{ b= }\sqrt[]{1}\text{ }\rightarrow\text{ b = 1} \end{gathered}$

For c,

$\text{ c = }\sqrt[]{a^2-b^2}\text{ = }\sqrt[]{(\frac{9}{7})^2\text{ - 1}}\text{ = }\sqrt[]{\frac{81}{49}\text{ - 1}}\text{ = }\sqrt[]{\frac{81}{49}-\frac{49}{49}}\text{ = }\sqrt[]{\frac{32}{49}}\text{ = }\frac{4\sqrt[]{2}}{7}$

Let's now get the foci of the ellipse:

The first focus: (h - c, k)

$\text{ (h - c, }k)\text{ = (0 - }\frac{4\sqrt[]{2}}{7},\text{ 0)}$$\text{ (h - c, }k)\text{ = (-}\frac{4\sqrt[]{2}}{7},\text{ 0)}$

The second focus: (h + c, k)

$\mleft(h+c,k\mright)\text{ = (0 + }\frac{4\sqrt[]{2}}{7},\text{ 0)}$$(h+c,k)\text{ = (}\frac{4\sqrt[]{2}}{7},\text{ 0)}$

Next, let's find the vertices:

The first vertex: (h − a, k)

$\text{ (h - a, k) = }(0\text{ - }\frac{9}{7},\text{ 0)}$$\text{ (h - a, k) = }(\text{-}\frac{9}{7},\text{ 0)}$

The second vertex: (h + a, k)

$\text{ (h + a, k) = }(0+\frac{9}{7},\text{ 0)}$$\text{ (h + a, k) = }(\frac{9}{7},\text{ 0)}$

In Summary:

Therefore, in an ellipse with the equation 49x^2 = 81 - 81y^2,

The foci of the ellipse are:

$\text{(-}\frac{4\sqrt[]{2}}{7},\text{ 0), (}\frac{4\sqrt[]{2}}{7},\text{ 0)}$

The vertices of the ellipse are:

$(\text{-}\frac{9}{7},\text{ 0), }(\frac{9}{7},\text{ 0)}$