12.1 The Ellipse

Complete the square for $x^{2} + 6 x$ to make it a perfect square.

Write the standard (graphing) form of the equation of the circle with radius 2 and center (-1, 3).

What is the standard form equation of the ellipse that has vertices $\left(\pm 8 , 0\right)$ and foci $\left(\pm 5 , 0\right) ?$

What is the standard form equation of the ellipse that has vertices $\left(0, \pm 4\right)$ and foci $\left(0, \pm \sqrt{15}\right) ?$

What is the standard form equation of the ellipse that has vertices $\left(−2 , −8\right)$ and $\left(−2 , \text{2}\right)$

and foci $\left(−2 , −7\right)$ and $\left(−2 , \text{1}\right) ?$

What is the standard form equation of the ellipse that has vertices $\left(−3 , 3\right)$ and $\left(5 , 3\right)$ and foci $\left(1 - 2 \sqrt{3} , 3\right)$ and $\left(1 + 2 \sqrt{3} , 3\right) ?$

Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.

Suppose a whispering chamber is 480 feet long and 320 feet wide.

ⓐ What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point $\left(0 , 0\right) .$
ⓑ If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot.

Define an ellipse in terms of its foci.

What special case of the ellipse do we have when the major and minor axis are of the same length?

What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the y-axis?

Determine whether the given equations represent ellipses. If yes, write in standard form.

$4 x^{2} + 9 y^{2} = 36$

Determine whether the given equations represent ellipses. If yes, write in standard form.

$4 x^{2} + 9 y^{2} = 1$

Write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

$\frac{x^{2}}{4} + \frac{y^{2}}{49} = 1$

Write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

$x^{2} + 9 y^{2} = 1$

Write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

$\frac{\left(x - 2\right)^{2}}{49} + \frac{\left(y - 4\right)^{2}}{25} = 1$

Write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

$\frac{\left(x + 5\right)^{2}}{4} + \frac{\left(y - 7\right)^{2}}{9} = 1$

Write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

$4 x^{2} - 8 x + 9 y^{2} - 72 y + 112 = 0$

Write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

$4 x^{2} - 24 x + 36 y^{2} - 360 y + 864 = 0$

Write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

$4 x^{2} + 40 x + 25 y^{2} - 100 y + 100 = 0$

Write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

$4 x^{2} + 24 x + 25 y^{2} + 200 y + 336 = 0$

Find the foci for the given ellipses.

$\frac{\left(x + 3\right)^{2}}{25} + \frac{\left(y + 1\right)^{2}}{36} = 1$

Find the foci for the given ellipses.

$x^{2} + y^{2} = 1$

Find the foci for the given ellipses.

$10 x^{2} + y^{2} + 200 x = 0$

Use the given information about the graph of each ellipse to determine its equation.

Center at the origin, symmetric with respect to the x- and y-axes, focus at $\left(0 , −2\right) ,$ and point on graph $\left(5 , 0\right) .$

Use the given information about the graph of each ellipse to determine its equation.

Center $\left(4 , 2\right)$ ; vertex $\left(9 , 2\right)$ ; one focus: $\left(4 + 2 \sqrt{6} , 2\right)$ .

Use the given information about the graph of each ellipse to determine its equation.

Center $\left(−3 , 4\right)$ ; vertex $\left(1 , 4\right)$ ; one focus: $\left(−3 + 2 \sqrt{3} , 4\right)$

Find the area of the ellipse. The area of an ellipse is given by the formula $\text{Area} = a \cdot b \cdot \pi .$

$\frac{\left(x - 3\right)^{2}}{9} + \frac{\left(y - 3\right)^{2}}{16} = 1$

Find the area of the ellipse. The area of an ellipse is given by the formula $\text{Area} = a \cdot b \cdot \pi .$

$\frac{\left(x + 1\right)^{2}}{4} + \frac{\left(y - 2\right)^{2}}{5} = 1$

Find the area of the ellipse. The area of an ellipse is given by the formula $\text{Area} = a \cdot b \cdot \pi .$

$9 x^{2} - 54 x + 9 y^{2} - 54 y + 81 = 0$

Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of $h ,$ the height.

An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.

A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.