Find the Properties 27(x-1)^2+36(y+2)^2=972

$27 {(x - 1)}^{2} + 36 {(y + 2)}^{2} = 972$

Step 1

Find the standard form of the ellipse.

Divide each term by $972$ to make the right side equal to one.

$\frac{27 {(x - 1)}^{2}}{972} + \frac{36 {(y + 2)}^{2}}{972} = \frac{972}{972}$

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 $1$ .

$\frac{{(x - 1)}^{2}}{36} + \frac{{(y + 2)}^{2}}{27} = 1$

Step 2

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.

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

Step 3

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.

$a = 6$

$b = 3 \sqrt{3}$

$k = - 2$

$h = 1$

Step 4

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

$(1, - 2)$

Step 5

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.

$\sqrt{{(6)}^{2} - {(3 \sqrt{3})}^{2}}$

Simplify.

Simplify the expression.

Raise $6$ to the power of $2$ .

$\sqrt{36 - {(3 \sqrt{3})}^{2}}$

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

$\sqrt{36 - (3^{2} {\sqrt{3}}^{2})}$

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

$\sqrt{36 - (9 {\sqrt{3}}^{2})}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{36 - (9 {(3^{\frac{1}{2}})}^{2})}$

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

$\sqrt{36 - (9 \cdot 3^{\frac{1}{2} \cdot 2})}$

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

$\sqrt{36 - (9 \cdot 3^{\frac{2}{2}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{36 - (9 \cdot 3^{\frac{2}{2}})}$

Rewrite the expression.

$\sqrt{36 - (9 \cdot 3^{1})}$

Evaluate the exponent.

$\sqrt{36 - (9 \cdot 3)}$

Multiply $- (9 \cdot 3)$ .

Multiply $9$ by $3$ .

$\sqrt{36 - 1 \cdot 27}$

Multiply $- 1$ by $27$ .

$\sqrt{36 - 27}$

Subtract $27$ from $36$ .

$\sqrt{9}$

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

$\sqrt{3^{2}}$

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

$3$

Step 6

Find the vertices.

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

$(h + a, k)$

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

$(1 + 6, - 2)$

Simplify.

$(7, - 2)$

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

$(h - a, k)$

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

$(1 - (6), - 2)$

Simplify.

$(- 5, - 2)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(7, - 2)$

${Vertex}_{2}$ : $(- 5, - 2)$

${Vertex}_{1}$ : $(7, - 2)$

${Vertex}_{2}$ : $(- 5, - 2)$

Step 7

Find the foci.

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

$(h + c, k)$

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

$(1 + 3, - 2)$

Simplify.

$(4, - 2)$

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

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

$(1 - (3), - 2)$

Simplify.

$(- 2, - 2)$

Ellipses have two foci.

${Focus}_{1}$ : $(4, - 2)$

${Focus}_{2}$ : $(- 2, - 2)$

${Focus}_{1}$ : $(4, - 2)$

${Focus}_{2}$ : $(- 2, - 2)$

Step 8

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.

$\frac{\sqrt{{(6)}^{2} - {(3 \sqrt{3})}^{2}}}{6}$

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$\frac{\sqrt{36 - {(3 \sqrt{3})}^{2}}}{6}$

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

$\frac{\sqrt{36 - (3^{2} {\sqrt{3}}^{2})}}{6}$

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

$\frac{\sqrt{36 - (9 {\sqrt{3}}^{2})}}{6}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{36 - (9 {(3^{\frac{1}{2}})}^{2})}}{6}$

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

$\frac{\sqrt{36 - (9 \cdot 3^{\frac{1}{2} \cdot 2})}}{6}$

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

$\frac{\sqrt{36 - (9 \cdot 3^{\frac{2}{2}})}}{6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{36 - (9 \cdot 3^{\frac{2}{2}})}}{6}$

Rewrite the expression.

$\frac{\sqrt{36 - (9 \cdot 3^{1})}}{6}$

Evaluate the exponent.

$\frac{\sqrt{36 - (9 \cdot 3)}}{6}$

Multiply $- (9 \cdot 3)$ .

Multiply $9$ by $3$ .

$\frac{\sqrt{36 - 1 \cdot 27}}{6}$

Multiply $- 1$ by $27$ .

$\frac{\sqrt{36 - 27}}{6}$

Subtract $27$ from $36$ .

$\frac{\sqrt{9}}{6}$

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

$\frac{\sqrt{3^{2}}}{6}$

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

$\frac{3}{6}$

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3$ .

$\frac{3 (1)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

$\frac{3 \cdot 1}{3 \cdot 2}$

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 2}$

Rewrite the expression.

$\frac{1}{2}$

Step 9

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

Center: $(1, - 2)$

${Vertex}_{1}$ : $(7, - 2)$

${Vertex}_{2}$ : $(- 5, - 2)$

${Focus}_{1}$ : $(4, - 2)$

${Focus}_{2}$ : $(- 2, - 2)$

Eccentricity: $\frac{1}{2}$

Step 10

Find the Properties 27(x-1)^2+36(y+2)^2=972

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