Problem 18 - College Algebra

Question

Graph each ellipse and locate the foci. $$6 x^{2}=30-5 y^{2}$$

Step-by-Step Solution

Answer

The foci for the given ellipse are at points (0,1) and (0,-1). The graph of the ellipse would be plotted such that it is vertically aligned with the foci at these points, and the semi-axes are \sqrt{5} (x-axis) and \sqrt{6} (y-axis).

Step 1: Rewrite the equation in standard form

Begin by writing the given equation, \(6x^2 = 30 - 5y^2\), in the standard form by dividing both sides by 30, yielding \(\frac{x^2}{5} - \frac{y^2}{6} = 1\). The constants 5 and 6 are the square of the semi-diameters.

Step 2: Recognize the type of the ellipse

To recognize the type of the ellipse, compare the coefficients. If \(a^2 < b^2\), then the major axis is along the y-axis. If \(a^2 > b^2\), then the major axis is along the x-axis. In our case, \(a^2 < b^2 (\sqrt{5} < \sqrt{6})\), thus, this is a vertical ellipse where a=\sqrt{6} and b=\sqrt{5}.

Step 3: Find the foci

The foci can now be found from the semi-axes using the formula \(f=\sqrt{a^2 - b^2}\). Substituting the values \(f=\sqrt{6 - 5}\) gives f= 1.

Step 4: Graph the ellipse and foci

To graph the ellipse, plot the center of the ellipse at the origin, then draw the ellipse using the semi-axes a and b. The foci are located on the major axis, 1 unit above and below the center. Plot the foci as two points on the y-axis, one at (0,1) and the other at (0,-1). Draw the ellipse around these points, making sure it intersects the x-axis at a=\sqrt{5} and -\sqrt{5}, and the y-axis at b=\sqrt{6} and -\sqrt{6}.