Find an equation for the ellipse. Graph the equation.
Center at left parenthesis 1 comma 1 right parenthesis; focus at left parenthesis 6 comma 1 right parenthesis; contains the point left parenthesis 1 comma 2 right parenthesis
The equation for an ellipse with center at (h,k), a horizontal major axis, and focal length 2c is:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
Given the center is (1,1) and the focus is (6,1), we know the distance from the center to the focus is 5 units. This means 2c = 5, so c = 5/2 = 2.5.
Since the ellipse contains the point (1,2), it must lie on the ellipse meaning 2.5 = sqrt(a^2 - b^2). So, a^2 = 2.5^2 + b^2.
To find the equation, we need to find the values of a and b:
2.5^2 = a^2 - b^2
6^2 = a^2
So, a^2 = 36, and b^2 = 11.25
Plugging these values into the equation of the ellipse, we get:
(x-1)^2/36 + (y-1)^2/11.25 = 1
This is the equation of the ellipse. Now, let's graph it:
The center is at (1,1), the major axis is along the x-axis with length 2a = 12, and the minor axis is along the y-axis with length 2b ≈ 6.7.
The graph will be an ellipse elongated horizontally with center at (1,1), focus at (6,1), and containing the point (1,2).
Unfortunately, as a text-based AI model, I cannot draw the graph for you. You can plot the points on graph paper or use a graphing tool to see the ellipse.