Find the Vertices 16x^2+25y^2-160x+150y+225=0

$16 x^{2} + 25 y^{2} - 160 x + 150 y + 225 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $225$ from both sides of the equation.

$16 x^{2} + 25 y^{2} - 160 x + 150 y = - 225$

Complete the square for $16 x^{2} - 160 x$ .

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

$a = 16$

$b = - 160$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 160}{2 \cdot 16}$

Simplify the right side.

Cancel the common factor of $- 160$ and $2$ .

Factor $2$ out of $- 160$ .

$d = \frac{2 \cdot - 80}{2 \cdot 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 16$ .

$d = \frac{2 \cdot - 80}{2 (16)}$

Cancel the common factor.

$d = \frac{2 \cdot - 80}{2 \cdot 16}$

Rewrite the expression.

$d = \frac{- 80}{16}$

Cancel the common factor of $- 80$ and $16$ .

Factor $16$ out of $- 80$ .

$d = \frac{16 \cdot - 5}{16}$

Cancel the common factors.

Factor $16$ out of $16$ .

$d = \frac{16 \cdot - 5}{16 (1)}$

Cancel the common factor.

$d = \frac{16 \cdot - 5}{16 \cdot 1}$

Rewrite the expression.

$d = \frac{- 5}{1}$

Divide $- 5$ by $1$ .

$d = - 5$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 160)}^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

Raise $- 160$ to the power of $2$ .

$e = 0 - \frac{25600}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = 0 - \frac{25600}{64}$

Divide $25600$ by $64$ .

$e = 0 - 1 \cdot 400$

Multiply $- 1$ by $400$ .

$e = 0 - 400$

Subtract $400$ from $0$ .

$e = - 400$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x - 5)}^{2} - 400$ .

$16 {(x - 5)}^{2} - 400$

Substitute $16 {(x - 5)}^{2} - 400$ for $16 x^{2} - 160 x$ in the equation $16 x^{2} + 25 y^{2} - 160 x + 150 y = - 225$ .

$16 {(x - 5)}^{2} - 400 + 25 y^{2} + 150 y = - 225$

Move $- 400$ to the right side of the equation by adding $400$ to both sides.

$16 {(x - 5)}^{2} + 25 y^{2} + 150 y = - 225 + 400$

Complete the square for $25 y^{2} + 150 y$ .

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

$a = 25$

$b = 150$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{150}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $150$ and $2$ .

Factor $2$ out of $150$ .

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Cancel the common factors.

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

$d = \frac{2 \cdot 75}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{75}{25}$

Cancel the common factor of $75$ and $25$ .

Factor $25$ out of $75$ .

$d = \frac{25 \cdot 3}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 3}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot 3}{25 \cdot 1}$

Rewrite the expression.

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{150^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $150$ to the power of $2$ .

$e = 0 - \frac{22500}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{22500}{100}$

Divide $22500$ by $100$ .

$e = 0 - 1 \cdot 225$

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Subtract $225$ from $0$ .

$e = - 225$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(y + 3)}^{2} - 225$ .

$25 {(y + 3)}^{2} - 225$

Substitute $25 {(y + 3)}^{2} - 225$ for $25 y^{2} + 150 y$ in the equation $16 x^{2} + 25 y^{2} - 160 x + 150 y = - 225$ .

$16 {(x - 5)}^{2} + 25 {(y + 3)}^{2} - 225 = - 225 + 400$

Move $- 225$ to the right side of the equation by adding $225$ to both sides.

$16 {(x - 5)}^{2} + 25 {(y + 3)}^{2} = - 225 + 400 + 225$

Simplify $- 225 + 400 + 225$ .

Add $- 225$ and $400$ .

$16 {(x - 5)}^{2} + 25 {(y + 3)}^{2} = 175 + 225$

Add $175$ and $225$ .

$16 {(x - 5)}^{2} + 25 {(y + 3)}^{2} = 400$

Divide each term by $400$ to make the right side equal to one.

$\frac{16 {(x - 5)}^{2}}{400} + \frac{25 {(y + 3)}^{2}}{400} = \frac{400}{400}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{25} + \frac{{(y + 3)}^{2}}{16} = 1$

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.

$a = 5$

$b = 4$

$k = - 3$

$h = 5$

Step 4

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.

$(5 + 5, - 3)$

Simplify.

$(10, - 3)$

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.

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

Simplify.

$(0, - 3)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(10, - 3)$

${Vertex}_{2}$ : $(0, - 3)$

${Vertex}_{1}$ : $(10, - 3)$

${Vertex}_{2}$ : $(0, - 3)$

Step 5

Find the Vertices 16x^2+25y^2-160x+150y+225=0

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