Find the Vertices 25x^2+4y^2-100x-8y+4=0

$25 x^{2} + 4 y^{2} - 100 x - 8 y + 4 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $4$ from both sides of the equation.

$25 x^{2} + 4 y^{2} - 100 x - 8 y = - 4$

Complete the square for $25 x^{2} - 100 x$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 25$

$b = - 100$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 100}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $- 100$ and $2$ .

Factor $2$ out of $- 100$ .

$d = \frac{2 \cdot - 50}{2 \cdot 25}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 25$ .

$d = \frac{2 \cdot - 50}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot - 50}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{- 50}{25}$

Cancel the common factor of $- 50$ and $25$ .

Factor $25$ out of $- 50$ .

$d = \frac{25 \cdot - 2}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot - 2}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot - 2}{25 \cdot 1}$

Rewrite the expression.

$d = \frac{- 2}{1}$

Divide $- 2$ by $1$ .

$d = - 2$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 100)}^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $- 100$ to the power of $2$ .

$e = 0 - \frac{10000}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{10000}{100}$

Divide $10000$ by $100$ .

$e = 0 - 1 \cdot 100$

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Subtract $100$ from $0$ .

$e = - 100$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(x - 2)}^{2} - 100$ .

$25 {(x - 2)}^{2} - 100$

Substitute $25 {(x - 2)}^{2} - 100$ for $25 x^{2} - 100 x$ in the equation $25 x^{2} + 4 y^{2} - 100 x - 8 y = - 4$ .

$25 {(x - 2)}^{2} - 100 + 4 y^{2} - 8 y = - 4$

Move $- 100$ to the right side of the equation by adding $100$ to both sides.

$25 {(x - 2)}^{2} + 4 y^{2} - 8 y = - 4 + 100$

Complete the square for $4 y^{2} - 8 y$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 4$

$b = - 8$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 8}{2 \cdot 4}$

Simplify the right side.

Cancel the common factor of $- 8$ and $2$ .

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 4}{2 (4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{- 4}{4}$

Cancel the common factor of $- 4$ and $4$ .

Factor $4$ out of $- 4$ .

$d = \frac{4 \cdot - 1}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$d = \frac{4 \cdot - 1}{4 (1)}$

Cancel the common factor.

$d = \frac{4 \cdot - 1}{4 \cdot 1}$

Rewrite the expression.

$d = \frac{- 1}{1}$

Divide $- 1$ by $1$ .

$d = - 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y - 1)}^{2} - 4$ .

$4 {(y - 1)}^{2} - 4$

Substitute $4 {(y - 1)}^{2} - 4$ for $4 y^{2} - 8 y$ in the equation $25 x^{2} + 4 y^{2} - 100 x - 8 y = - 4$ .

$25 {(x - 2)}^{2} + 4 {(y - 1)}^{2} - 4 = - 4 + 100$

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

$25 {(x - 2)}^{2} + 4 {(y - 1)}^{2} = - 4 + 100 + 4$

Simplify $- 4 + 100 + 4$ .

Add $- 4$ and $100$ .

$25 {(x - 2)}^{2} + 4 {(y - 1)}^{2} = 96 + 4$

Add $96$ and $4$ .

$25 {(x - 2)}^{2} + 4 {(y - 1)}^{2} = 100$

Divide each term by $100$ to make the right side equal to one.

$\frac{25 {(x - 2)}^{2}}{100} + \frac{4 {(y - 1)}^{2}}{100} = \frac{100}{100}$

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 $1$ .

$\frac{{(x - 2)}^{2}}{4} + \frac{{(y - 1)}^{2}}{25} = 1$

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.

$a = 5$

$b = 2$

$k = 1$

$h = 2$

Step 4

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.

$(2, 1 + 5)$

Simplify.

$(2, 6)$

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.

$(2, 1 - (5))$

Simplify.

$(2, - 4)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(2, 6)$

${Vertex}_{2}$ : $(2, - 4)$

${Vertex}_{1}$ : $(2, 6)$

${Vertex}_{2}$ : $(2, - 4)$

Step 5

Find the Vertices 25x^2+4y^2-100x-8y+4=0

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