Problem 74 - Algebra and Trigonometry

Question

Describe how to graph \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Step-by-Step Solution

Answer

The graph of the given equation is an ellipse centered at the origin (0,0) with a horizontal radius of 5 units, a vertical radius of 4 units, and vertices at (-5,0), (5,0), (0,-4), and (0,4). This is evident in the plotted graph.

Step 1: Identifying the center, the horizontal radius and the vertical radius

Given the equation \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\), notice that it is in the standard form of ellipse equations \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). This type of ellipse is centered at the origin (0,0), with a horizontal radius 'a' of 5 units (found by taking the square root of the denominator under the x-term) and a vertical radius 'b' of 4 units (found by taking the square root of the denominator under the y-term). Therefore, the center is (0, 0); the horizontal radius is 5 units, and the vertical radius is 4 units.

Step 2: Finding the vertices of the ellipse

The vertices of this ellipse can be found using the radii 'a' and 'b'. Because the horizontal radius is 5, and the vertical radius is 4, we have two vertices at (-5, 0) and (5, 0) along the x-axis and two vertices at (0, -4) and (0, 4) along the y-axis.

Step 3: Plotting the ellipse

Using the determined center, radii, and vertices, the ellipse can be sketched. Begin by plotting the center point (0,0). Plot the x-axis vertices at (-5,0) and (5,0). Also plot the y-axis vertices at (0,-4) and (0,4). Draw an oval-like curve that touches these points to complete the graph of the ellipse.