Question

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph.

3x^2 + y^2 + 18x - 2y - 8 = 0

Answer 1

Step 1:

The conic equation is

The coefficients of and are the same sign but unequal coefficients. So the equation is ellipse.

Convert the equation in standard form of ellipse.

To change the expressions and into a perfect square trinomial,

add (half the x coefficient)² and add (half the y coefficient)² to each side of the equation.

Compare it to standard form of vertical ellipse is .

Where

a  is length of semi major axis and b is length of semi minor axis.

Center is , vertices

Foci and .

Where .

In this case .

.

Vertices are .

Foci .

Eccentricity .

.

Step 2:

Graph:

Draw the coordinate plane.

Plot the center, vertices and foci of ellipse.

Then sketch the ellipse, use the semi major axis length is 6 units and semi minor axis length is 3.46 units.

Solution:

Center , vertices .

Foci , .