Find the Properties 49x^2+16y^2=784

$49 x^{2} + 16 y^{2} = 784$

Find the standard form of the ellipse.

Divide each term by $784$ to make the right side equal to one.

$\frac{49 x^{2}}{784} + \frac{16 y^{2}}{784} = \frac{784}{784}$

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{x^{2}}{16} + \frac{y^{2}}{49} = 1$

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

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

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = 7$

$b = 4$

$k = 0$

$h = 0$

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 0)$

Find $c$ , the distance from the center to a focus.

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(7)}^{2} - {(4)}^{2}}$

Simplify.

Raise $7$ to the power of $2$ .

$\sqrt{49 - {(4)}^{2}}$

Raise $4$ to the power of $2$ .

$\sqrt{49 - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

$\sqrt{49 - 16}$

Subtract $16$ from $49$ .

$\sqrt{33}$

Find the vertices.

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(0, 0 + 7)$

Simplify.

$(0, 7)$

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(0, 0 - (7))$

Simplify.

$(0, - 7)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 7)$

${Vertex}_{2}$ : $(0, - 7)$

${Vertex}_{1}$ : $(0, 7)$

${Vertex}_{2}$ : $(0, - 7)$

Find the foci.

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0, 0 + \sqrt{33})$

Simplify.

$(0, \sqrt{33})$

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0, 0 - (\sqrt{33}))$

Simplify.

$(0, - \sqrt{33})$

Ellipses have two foci.

${Focus}_{1}$ : $(0, \sqrt{33})$

${Focus}_{2}$ : $(0, - \sqrt{33})$

${Focus}_{1}$ : $(0, \sqrt{33})$

${Focus}_{2}$ : $(0, - \sqrt{33})$

Find the eccentricity.

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(7)}^{2} - {(4)}^{2}}}{7}$

Simplify the numerator.

Raise $7$ to the power of $2$ .

$\frac{\sqrt{49 - 4^{2}}}{7}$

Raise $4$ to the power of $2$ .

$\frac{\sqrt{49 - 1 \cdot 16}}{7}$

Multiply $- 1$ by $16$ .

$\frac{\sqrt{49 - 16}}{7}$

Subtract $16$ from $49$ .

$\frac{\sqrt{33}}{7}$

These values represent the important values for graphing and analyzing an ellipse.

Center: $(0, 0)$

${Vertex}_{1}$ : $(0, 7)$

${Vertex}_{2}$ : $(0, - 7)$

${Focus}_{1}$ : $(0, \sqrt{33})$

${Focus}_{2}$ : $(0, - \sqrt{33})$

Eccentricity: $\frac{\sqrt{33}}{7}$

Find the Properties 49x^2+16y^2=784

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