Find the Foci 7(x-2)^2+3(y-2)^2=21

$7 {(x - 2)}^{2} + 3 {(y - 2)}^{2} = 21$

Step 1

Find the standard form of the ellipse.

Divide each term by $21$ to make the right side equal to one.

$\frac{7 {(x - 2)}^{2}}{21} + \frac{3 {(y - 2)}^{2}}{21} = \frac{21}{21}$

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 $1$ .

$\frac{{(x - 2)}^{2}}{3} + \frac{{(y - 2)}^{2}}{7} = 1$

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.

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

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.

$a = \sqrt{7}$

$b = \sqrt{3}$

$k = 2$

$h = 2$

Step 4

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.

$\sqrt{{(\sqrt{7})}^{2} - {(\sqrt{3})}^{2}}$

Simplify.

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{{(7^{\frac{1}{2}})}^{2} - {(\sqrt{3})}^{2}}$

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

$\sqrt{7^{\frac{1}{2} \cdot 2} - {(\sqrt{3})}^{2}}$

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

$\sqrt{7^{\frac{2}{2}} - {(\sqrt{3})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{7^{\frac{2}{2}} - {(\sqrt{3})}^{2}}$

Rewrite the expression.

$\sqrt{7^{1} - {(\sqrt{3})}^{2}}$

Evaluate the exponent.

$\sqrt{7 - {(\sqrt{3})}^{2}}$

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

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

$\sqrt{7 - {(3^{\frac{1}{2}})}^{2}}$

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

$\sqrt{7 - 3^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{7 - 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{7 - 3^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{7 - 3^{1}}$

Evaluate the exponent.

$\sqrt{7 - 1 \cdot 3}$

Simplify the expression.

Multiply $- 1$ by $3$ .

$\sqrt{7 - 3}$

Subtract $3$ from $7$ .

$\sqrt{4}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2}}$

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

$2$

Step 5

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.

$(2, 2 + 2)$

Simplify.

$(2, 4)$

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.

$(2, 2 - (2))$

Simplify.

$(2, 0)$

Ellipses have two foci.

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

${Focus}_{2}$ : $(2, 0)$

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

${Focus}_{2}$ : $(2, 0)$

Step 6

Find the Foci 7(x-2)^2+3(y-2)^2=21

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