Problem 39 - College Algebra

Question

Graph each ellipse and give the location of its foci. $$(x+3)^{2}+4(y-2)^{2}=16$$

Step-by-Step Solution

Answer

The center of the ellipse is at \((-3,2)\), \(a = 4\) and \(b = 2\). The location of the foci are \((-3-2\sqrt{3}, 2)\) and \((-3+2\sqrt{3}, 2)\).

Step 1: Identifying the Center, a, and b

The general equation of an ellipse in the form \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\) where the center of the ellipse is \((h, k)\), and \(a\) and \(b\) are the semi-major and semi-minor axes respectively. From the given equation \((x+3)^{2}+4(y-2)^{2}=16\), by comparing with the general equation, the center \((h,k)\) of the ellipse is at \((-3,2)\). The values of \(a^2\) and \(b^2\) can be determined by equating them to the coefficients of \(x^2\) and \(y^2\) respectively. So \(a^2 = 16\) and \(4b^2 = 16\). Solving for \(a\) and \(b\), we get \(a = 4\) and \(b = 2\).

Step 2: Calculating the Foci

To calculate the foci, use the formula \(c= \sqrt{a^2 - b^2}\). Plugging in the values obtained in step 1, The calculation becomes \(c= \sqrt{4^2 - 2^2} = 2\sqrt{3}\). The foci will be \(c\) unit distance from the center along the major axis. As the value of \(a^2\) is under \(x^2\), the major axis is along the x-axis. So, the coordinates of the foci will be \((-3-2\sqrt{3}, 2)\) and \((-3+2\sqrt{3}, 2)\).

Step 3: Plotting the Ellipse and the Foci

Now, plot the ellipse using its center, \(a\) and \(b\). The h and k in ellipse equation represents the shift of the ellipse along the x and y-axis respectively. The signs of h and k are reversed in the ellipse equation. Hence, move 3 units left and 2 units up from the origin to plot the center of the ellipse. Then sketch the boundaries of the ellipse by moving a and b units along the x and y axis respectively and connect them smoothly to form the ellipse. Afterwards, plot the foci that were calculated in the second step. Drawing all these gives a complete plot of the ellipse and its foci.