Problem 49 - College Algebra

Question

Graph each ellipse and give the location of its foci. $$9(x-1)^{2}+4(y+3)^{2}=36$$

Step-by-Step Solution

Answer

The ellipse has its center at (1, -3), its foci at (1 − √5, -3) and (1 + √5, -3), and it is vertically oriented with a semi-major axis of length 3 and a semi-minor axis of length 2.

Step 1: Simplify the Equation

First modify the equation \( 9(x-1)^{2}+4(y+3)^{2}=36 \) into generalized form by dividing every term by 36. This results in \( \frac{(x-1)^2}{4} + \frac{(y+3)^2}{9} = 1 \)

Step 2: Identify Center, a and b

From the simplified equation, identify the center, a (half-length of the major axis) and b (half-length of the minor axis). Here, the center is (h, k) = (1, -3), a = 3, and b = 2.

Step 3: Compute the Foci

Use the relationship \( c = \sqrt{a^{2}-b^{2}} \) to find the foci. Here, \( c = \sqrt{3^{2}-2^{2}} = \sqrt{5} \), thus the foci are at (h ± c, k) = (1 − √5, -3) and (1 + √5, -3).

Step 4: Graph the Ellipse

You start by plotting the center of the ellipse. Then, plot the vertices which are 'a' units above and below the center (since a>b, the ellipse is vertically oriented). After that, draw the ellipse as best you can through these points. Lastly, plot the foci within the ellipse. The foci are 'c' units above and below the center.