# Ellipses: Standard Form - Video Tutorials & Practice Problems

## Graph Ellipses at Origin

Given the equation $\frac{x^2}{4}+\frac{y^2}{9}=1$, sketch a graph of the ellipse.

Given the ellipse equation $\frac{x^2}{16}+\frac{y^2}{4}=1$, determine the magnitude of the semi-major axis (a) and the semi-minor axis (b).

$a=16$, $b=4$

$a=4$, $b=16$

$a=4$, $b=2$

$a=2$, $b=4$

## Foci and Vertices of an Ellipse

Determine the vertices and foci of the following ellipse: $\frac{x^2}{49}+\frac{y^2}{36}=1$.

Vertices: $\left(7,0\right),\left(-7,0\right)$

Foci: $\left(6,0\right),\left(-6,0\right)$

Vertices: $\left(6,0\right),\left(-6,0\right)$

Foci: $\left(7,0\right),\left(-7,0\right)$

Vertices: $\left(7,0\right),\left(-7,0\right)$

Foci: $\left(\sqrt{13},0\right),\left(-\sqrt{13},0\right)$

Vertices: $\left(0,7\right),\left(0,-7\right)$

Foci: $\left(0,\sqrt{13}\right),\left(0,-\sqrt{13}\right)$

Determine the vertices and foci of the following ellipse: $\frac{x^2}{9}+\frac{y^2}{16}=1$.

Vertices: $\left(4,0\right),\left(-4,0\right)$

Foci: $\left(\sqrt7,0\right),\left(-\sqrt7,0\right)$

Vertices: $\left(0,4\right),\left(0,-4\right)$

Foci: $\left(0,\sqrt7\right),\left(0,-\sqrt7\right)$

Vertices: $\left(4,0\right),\left(-4,0\right)$

Foci: $\left(3,0\right),\left(-3,0\right)$

Vertices: $\left(0,4\right),\left(0,-4\right)$

Foci: $\left(0,3\right),\left(0,-3\right)$

Find the standard form of the equation for an ellipse with the following conditions.

Foci = $\left(-5,0\right),\left(5,0\right)$

Vertices = $\left(-8,0\right),\left(8,0\right)$

$\frac{x^2}{64}+\frac{y^2}{25}=1$

$\frac{x^2}{25}+\frac{y^2}{64}=1$

$\frac{x^2}{8}+\frac{y^2}{5}=1$

$\frac{x^2}{64}+\frac{y^2}{39}=1$

## Graph Ellipses NOT at Origin

Graph the ellipse $\frac{{(x-1)}^{2}}{9}+\frac{{(y+3)}^{2}}{4}=1$.

Determine the vertices and foci of the ellipse $\left(x+1\right)^2+\frac{\left(y-2\right)^2}{4}=1$.

Vertices: $\left(-1,4\right),\left(-1,0\right)$

Foci: $\left(-1,2+\sqrt3\right),\left(-1,2-\sqrt3\right)$

Vertices: $\left(-1,4\right),\left(-1,0\right)$

Foci: $\left(-2,2\right),\left(0,2\right)$

Vertices: $\left(-2,2\right),\left(0,2\right)$

Foci: $\left(1,2+\sqrt3\right),\left(1,2-\sqrt3\right)$

Vertices: $\left(-2,2\right),\left(0,2\right)$

Foci: $\left(2+\sqrt3,1\right),\left(2-\sqrt3,1\right)$

## Do you want more practice?

- Graph the ellipse and locate the foci. (y^2)/25 + (x^2)/16 = 1
- Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (-4,0), (4,0); Ve...
- Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis horizontal w...
- Graph the ellipse and locate the foci. 9x^2 + 4y^2 - 18x + 8y -23 = 0
- Graph the ellipse and locate the foci. (y^2)/25 + (x^2)/16 = 1
- Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis horizontal w...
- In Exercises 1–18, graph each ellipse and locate the foci. x^2/16 +y^2/4 = 1
- Graph the ellipse and locate the foci. (x^2)/36 +(y^2)/25 = 1
- Graph the ellipse and locate the foci. (x^2)/36 +(y^2)/25 = 1
- In Exercises 1–18, graph each ellipse and locate the foci. x^2/9 +y^2/36= 1
- In Exercises 1–18, graph each ellipse and locate the foci. x^2/25 +y^2/64 = 1
- In Exercises 1–18, graph each ellipse and locate the foci. x^2/49 +y^2/81 = 1
- In Exercises 1–18, graph each ellipse and locate the foci. x^2/49 +y^2/81 = 1
- In Exercises 1–18, graph each ellipse and locate the foci. x^2/(9/4) +y^2/(25/4) = 1
- In Exercises 1–18, graph each ellipse and locate the foci. x^2/(9/4) +y^2/(25/4) = 1
- In Exercises 1–18, graph each ellipse and locate the foci. x² = 1 – 4y²
- In Exercises 1–18, graph each ellipse and locate the foci. 25x²+4y² = 100
- In Exercises 1–18, graph each ellipse and locate the foci.4x²+16y² = 64
- In Exercises 1–18, graph each ellipse and locate the foci.4x²+16y² = 64
- In Exercises 1–18, graph each ellipse and locate the foci. 7x² = 35-5y²
- In Exercises 1–18, graph each ellipse and locate the foci. 7x² = 35-5y²
- In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.
- In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.
- In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.
- In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Fo...
- In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Fo...
- In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Fo...
- In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Ma...
- In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Ma...
- In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Ma...
- In Exercises 37–50, graph each ellipse and give the location of its foci. (x − 2)²/9 + (y -1)² /4= 1
- In Exercises 37–50, graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16
- In Exercises 37–50, graph each ellipse and give the location of its foci. (x − 4)²/9 + (y +2)² /25= 1
- In Exercises 37–50, graph each ellipse and give the location of its foci. x²/25 + (y -2)² /36= 1
- In Exercises 37–50, graph each ellipse and give the location of its foci. (x +3)²/9 + (y -2)² = 1
- In Exercises 37–50, graph each ellipse and give the location of its foci. (x − 1)²/2 + (y +3)² /5= 1
- In Exercises 37–50, graph each ellipse and give the location of its foci. 9(x − 1)²+4(y+3)² = 36
- In Exercises 49–56, identify each equation without completing the square. 4x^2 - 9y^2 - 8x - 36y - 68 = 0
- In Exercises 51–60, convert each equation to standard form by completing the square on x and y. Then graph the...
- In Exercises 51–60, convert each equation to standard form by completing the square on x and y. Then graph the...
- In Exercises 51–60, convert each equation to standard form by completing the square on x and y. Then graph the...
- In Exercises 61–66, find the solution set for each system by graphing both of the system's equations in the sa...
- In Exercises 61–66, find the solution set for each system by graphing both of the system's equations in the sa...
- In Exercises 61–66, find the solution set for each system by graphing both of the system's equations in the sa...
- In Exercises 67–68, graph each semiellipse. y = √16 - 4x²
- Find the standard form of the equation of an ellipse with vertices at (0, -6) and (0, 6), passing through (2,...
- The equation of the red ellipse in the figure shown is x^2/25 + y^2/9 =1Write the equation for each circle sho...