Find the Ellipse: Center (-1,2.2), Focus (-1,2), Vertex (-1,1.22) (-1,2.2) , (-1,1.22) , (-1,2)

$(- 1, 2.2)$ , $(- 1, 1.22)$ , $(- 1, 2)$

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 $(- 1, 1.22)$ and the center point $(- 1, 2.2)$ .

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.

$a = \sqrt{{((- 1) - (- 1))}^{2} + {(1.22 - 2.2)}^{2}}$

Simplify.

Multiply $- 1$ by $- 1$ .

$a = \sqrt{{(- 1 + 1)}^{2} + {(1.22 - 2.2)}^{2}}$

Add $- 1$ and $1$ .

$a = \sqrt{0^{2} + {(1.22 - 2.2)}^{2}}$

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

$a = \sqrt{0 + {(1.22 - 2.2)}^{2}}$

Subtract $2.2$ from $1.22$ .

$a = \sqrt{0 + {(- 0.98)}^{2}}$

Raise $- 0.98$ to the power of $2$ .

$a = \sqrt{0 + 0.9604}$

Add $0$ and $0.9604$ .

$a = \sqrt{0.9604}$

Step 3

$c$ is the distance between the focus $(- 1, 2)$ and the center $(- 1, 2.2)$ .

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.

$c = \sqrt{{((- 1) - (- 1))}^{2} + {(2 - 2.2)}^{2}}$

Simplify.

Multiply $- 1$ by $- 1$ .

$c = \sqrt{{(- 1 + 1)}^{2} + {(2 - 2.2)}^{2}}$

Add $- 1$ and $1$ .

$c = \sqrt{0^{2} + {(2 - 2.2)}^{2}}$

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

$c = \sqrt{0 + {(2 - 2.2)}^{2}}$

Subtract $2.2$ from $2$ .

$c = \sqrt{0 + {(- 0.2)}^{2}}$

Raise $- 0.2$ to the power of $2$ .

$c = \sqrt{0 + 0.04}$

Add $0$ and $0.04$ .

$c = \sqrt{0.04}$

Rewrite $0.04$ as ${0.2}^{2}$ .

$c = \sqrt{{0.2}^{2}}$

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

$c = 0.2$

Step 4

Using the equation $c^{2} = a^{2} - b^{2}$ . Substitute $\sqrt{0.9604}$ for $a$ and $0.2$ for $c$ .

Rewrite the equation as ${(\sqrt{0.9604})}^{2} - b^{2} = {0.2}^{2}$ .

${(\sqrt{0.9604})}^{2} - b^{2} = {0.2}^{2}$

Rewrite ${\sqrt{0.9604}}^{2}$ as $0.9604$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{0.9604}$ as ${0.9604}^{\frac{1}{2}}$ .

${({0.9604}^{\frac{1}{2}})}^{2} - b^{2} = {0.2}^{2}$

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

${0.9604}^{\frac{1}{2} \cdot 2} - b^{2} = {0.2}^{2}$

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

${0.9604}^{\frac{2}{2}} - b^{2} = {0.2}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${0.9604}^{\frac{2}{2}} - b^{2} = {0.2}^{2}$

Rewrite the expression.

${0.9604}^{1} - b^{2} = {0.2}^{2}$

Evaluate the exponent.

$0.9604 - b^{2} = {0.2}^{2}$

Raise $0.2$ to the power of $2$ .

$0.9604 - b^{2} = 0.04$

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

Subtract $0.9604$ from both sides of the equation.

$- b^{2} = 0.04 - 0.9604$

Subtract $0.9604$ from $0.04$ .

$- b^{2} = - 0.9204$

Divide each term in $- b^{2} = - 0.9204$ by $- 1$ and simplify.

Divide each term in $- b^{2} = - 0.9204$ by $- 1$ .

$\frac{- b^{2}}{- 1} = \frac{- 0.9204}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{b^{2}}{1} = \frac{- 0.9204}{- 1}$

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

$b^{2} = \frac{- 0.9204}{- 1}$

Simplify the right side.

Divide $- 0.9204$ by $- 1$ .

$b^{2} = 0.9204$

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

$b = \pm \sqrt{0.9204}$

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.

$b = \sqrt{0.9204}$

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

$b = - \sqrt{0.9204}$

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

$b = \sqrt{0.9204}, - \sqrt{0.9204}$

Step 5

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

$b = \sqrt{0.9204}$

Step 6

The slope of the line between the focus $(- 1, 2)$ and the center $(- 1, 2.2)$ 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.

$m = \frac{2.2 - (2)}{- 1 - (- 1)}$

Simplify.

Multiply $- 1$ by $- 1$ .

$\frac{2.2 - (2)}{- 1 + 1}$

Add $- 1$ and $1$ .

$\frac{2.2 - (2)}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

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

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

Step 7

Substitute the values $h = - 1$ , $k = 2.2$ , $a = \sqrt{0.9604}$ , and $b = \sqrt{0.9204}$ into $\frac{{(y - k)}^{2}}{a^{2}} + \frac{{(x - h)}^{2}}{b^{2}} = 1$ to get the ellipse equation $\frac{{(y - (2.2))}^{2}}{{(\sqrt{0.9604})}^{2}} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$ .

$\frac{{(y - (2.2))}^{2}}{{(\sqrt{0.9604})}^{2}} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Step 8

Simplify to find the final equation of the ellipse.

Multiply $- 1$ by $2.2$ .

$\frac{{(y - 2.2)}^{2}}{{\sqrt{0.9604}}^{2}} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Rewrite ${\sqrt{0.9604}}^{2}$ as $0.9604$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{0.9604}$ as ${0.9604}^{\frac{1}{2}}$ .

$\frac{{(y - 2.2)}^{2}}{{({0.9604}^{\frac{1}{2}})}^{2}} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

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

$\frac{{(y - 2.2)}^{2}}{{0.9604}^{\frac{1}{2} \cdot 2}} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

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

$\frac{{(y - 2.2)}^{2}}{{0.9604}^{\frac{2}{2}}} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{{(y - 2.2)}^{2}}{{0.9604}^{\frac{2}{2}}} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Rewrite the expression.

$\frac{{(y - 2.2)}^{2}}{0.9604} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Evaluate the exponent.

$\frac{{(y - 2.2)}^{2}}{0.9604} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Multiply by $1$ .

$\frac{1 {(y - 2.2)}^{2}}{0.9604} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Factor $0.9604$ out of $0.9604$ .

$\frac{1 {(y - 2.2)}^{2}}{0.9604 (1)} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Separate fractions.

$\frac{1}{0.9604} \cdot \frac{{(y - 2.2)}^{2}}{1} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Divide $1$ by $0.9604$ .

$1.04123281 (\frac{{(y - 2.2)}^{2}}{1}) + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Divide ${(y - 2.2)}^{2}$ by $1$ .

$1.04123281 {(y - 2.2)}^{2} + \frac{{(x - (- 1))}^{2}}{{(\sqrt{0.9204})}^{2}} = 1$

Multiply $- 1$ by $- 1$ .

$1.04123281 {(y - 2.2)}^{2} + \frac{{(x + 1)}^{2}}{{\sqrt{0.9204}}^{2}} = 1$

Rewrite ${\sqrt{0.9204}}^{2}$ as $0.9204$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{0.9204}$ as ${0.9204}^{\frac{1}{2}}$ .

$1.04123281 {(y - 2.2)}^{2} + \frac{{(x + 1)}^{2}}{{({0.9204}^{\frac{1}{2}})}^{2}} = 1$

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

$1.04123281 {(y - 2.2)}^{2} + \frac{{(x + 1)}^{2}}{{0.9204}^{\frac{1}{2} \cdot 2}} = 1$

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

$1.04123281 {(y - 2.2)}^{2} + \frac{{(x + 1)}^{2}}{{0.9204}^{\frac{2}{2}}} = 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$1.04123281 {(y - 2.2)}^{2} + \frac{{(x + 1)}^{2}}{{0.9204}^{\frac{2}{2}}} = 1$

Rewrite the expression.

$1.04123281 {(y - 2.2)}^{2} + \frac{{(x + 1)}^{2}}{0.9204} = 1$

Evaluate the exponent.

$1.04123281 {(y - 2.2)}^{2} + \frac{{(x + 1)}^{2}}{0.9204} = 1$

Multiply by $1$ .

$1.04123281 {(y - 2.2)}^{2} + \frac{1 {(x + 1)}^{2}}{0.9204} = 1$

Factor $0.9204$ out of $0.9204$ .

$1.04123281 {(y - 2.2)}^{2} + \frac{1 {(x + 1)}^{2}}{0.9204 (1)} = 1$

Separate fractions.

$1.04123281 {(y - 2.2)}^{2} + \frac{1}{0.9204} \cdot \frac{{(x + 1)}^{2}}{1} = 1$

Divide $1$ by $0.9204$ .

$1.04123281 {(y - 2.2)}^{2} + 1.08648413 (\frac{{(x + 1)}^{2}}{1}) = 1$

Divide ${(x + 1)}^{2}$ by $1$ .

$1.04123281 {(y - 2.2)}^{2} + 1.08648413 {(x + 1)}^{2} = 1$

Step 9

Find the Ellipse: Center (-1,2.2), Focus (-1,2), Vertex (-1,1.22) (-1,2.2) , (-1,1.22) , (-1,2)

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