Find the Ellipse: Center (5,3), Focus (4,3), Vertex (-5,3) (5,3) , (4,3) , (-5,3)

$(5, 3)$ , $(4, 3)$ , $(- 5, 3)$

Step 1

There are two general equations for an ellipse.

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

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

Step 2

$a$ is the distance between the vertex $(- 5, 3)$ and the center point $(5, 3)$ .

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Substitute the actual values of the points into the distance formula.

$a = \sqrt{{((- 5) - 5)}^{2} + {(3 - 3)}^{2}}$

Simplify.

Subtract $5$ from $- 5$ .

$a = \sqrt{{(- 10)}^{2} + {(3 - 3)}^{2}}$

Raise $- 10$ to the power of $2$ .

$a = \sqrt{100 + {(3 - 3)}^{2}}$

Subtract $3$ from $3$ .

$a = \sqrt{100 + 0^{2}}$

Raising $0$ to any positive power yields $0$ .

$a = \sqrt{100 + 0}$

Add $100$ and $0$ .

$a = \sqrt{100}$

Rewrite $100$ as $10^{2}$ .

$a = \sqrt{10^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$a = 10$

Step 3

$c$ is the distance between the focus $(4, 3)$ and the center $(5, 3)$ .

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Substitute the actual values of the points into the distance formula.

$c = \sqrt{{(4 - 5)}^{2} + {(3 - 3)}^{2}}$

Simplify.

Subtract $5$ from $4$ .

$c = \sqrt{{(- 1)}^{2} + {(3 - 3)}^{2}}$

Raise $- 1$ to the power of $2$ .

$c = \sqrt{1 + {(3 - 3)}^{2}}$

Subtract $3$ from $3$ .

$c = \sqrt{1 + 0^{2}}$

Raising $0$ to any positive power yields $0$ .

$c = \sqrt{1 + 0}$

Add $1$ and $0$ .

$c = \sqrt{1}$

Any root of $1$ is $1$ .

$c = 1$

Step 4

Using the equation $c^{2} = a^{2} - b^{2}$ . Substitute $10$ for $a$ and $1$ for $c$ .

Rewrite the equation as ${(10)}^{2} - b^{2} = 1^{2}$ .

${(10)}^{2} - b^{2} = 1^{2}$

Raise $10$ to the power of $2$ .

$100 - b^{2} = 1^{2}$

One to any power is one.

$100 - b^{2} = 1$

Move all terms not containing $b$ to the right side of the equation.

Subtract $100$ from both sides of the equation.

$- b^{2} = 1 - 100$

Subtract $100$ from $1$ .

$- b^{2} = - 99$

Divide each term in $- b^{2} = - 99$ by $- 1$ and simplify.

Divide each term in $- b^{2} = - 99$ by $- 1$ .

$\frac{- b^{2}}{- 1} = \frac{- 99}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{b^{2}}{1} = \frac{- 99}{- 1}$

Divide $b^{2}$ by $1$ .

$b^{2} = \frac{- 99}{- 1}$

Simplify the right side.

Divide $- 99$ by $- 1$ .

$b^{2} = 99$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$b = \pm \sqrt{99}$

Simplify $\pm \sqrt{99}$ .

Rewrite $99$ as $3^{2} \cdot 11$ .

Factor $9$ out of $99$ .

$b = \pm \sqrt{9 (11)}$

Rewrite $9$ as $3^{2}$ .

$b = \pm \sqrt{3^{2} \cdot 11}$

Pull terms out from under the radical.

$b = \pm 3 \sqrt{11}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$b = 3 \sqrt{11}$

Next, use the negative value of the $\pm$ to find the second solution.

$b = - 3 \sqrt{11}$

The complete solution is the result of both the positive and negative portions of the solution.

$b = 3 \sqrt{11}, - 3 \sqrt{11}$

Step 5

$b$ is a distance, which means it should be a positive number.

$b = 3 \sqrt{11}$

Step 6

The slope of the line between the focus $(4, 3)$ and the center $(5, 3)$ determines whether the ellipse is vertical or horizontal. If the slope is $0$ , the graph is horizontal. If the slope is undefined, the graph is vertical.

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{3 - (3)}{5 - (4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{3 - 3}{5 - (4)}$

Subtract $3$ from $3$ .

$m = \frac{0}{5 - (4)}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$m = \frac{0}{5 - 4}$

Subtract $4$ from $5$ .

$m = \frac{0}{1}$

Divide $0$ by $1$ .

$m = 0$

The general equation for a horizontal ellipse is $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$ .

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

Step 7

Substitute the values $h = 5$ , $k = 3$ , $a = 10$ , and $b = 3 \sqrt{11}$ into $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$ to get the ellipse equation $\frac{{(x - (5))}^{2}}{{(10)}^{2}} + \frac{{(y - (3))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$ .

$\frac{{(x - (5))}^{2}}{{(10)}^{2}} + \frac{{(y - (3))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Step 8

Simplify to find the final equation of the ellipse.

Multiply $- 1$ by $5$ .

$\frac{{(x - 5)}^{2}}{10^{2}} + \frac{{(y - (3))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Raise $10$ to the power of $2$ .

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - (3))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Multiply $- 1$ by $3$ .

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Simplify the denominator.

Apply the product rule to $3 \sqrt{11}$ .

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{3^{2} {\sqrt{11}}^{2}} = 1$

Raise $3$ to the power of $2$ .

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{9 {\sqrt{11}}^{2}} = 1$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{9 {(11^{\frac{1}{2}})}^{2}} = 1$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{9 \cdot 11^{\frac{1}{2} \cdot 2}} = 1$

Combine $\frac{1}{2}$ and $2$ .

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{9 \cdot 11^{\frac{2}{2}}} = 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{9 \cdot 11^{\frac{2}{2}}} = 1$

Rewrite the expression.

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{9 \cdot 11} = 1$

Evaluate the exponent.

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{9 \cdot 11} = 1$

Multiply $9$ by $11$ .

$\frac{{(x - 5)}^{2}}{100} + \frac{{(y - 3)}^{2}}{99} = 1$

Step 9

Find the Ellipse: Center (5,3), Focus (4,3), Vertex (-5,3) (5,3) , (4,3) , (-5,3)

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