1. Describe the graph of an ellipse centered at the origin given the following conditions:a. a>bb. a<bc. a=b2. As you consider the shape of the graph of an ellipse where a = b, can you consider a circle as an ellipse? Why or why not?3. Describe the coordinates of the x-intercepts and y-intercepts of an ellipse centered at the origin.
Answer (1)
Answer: I'll use the standard form\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1Step-by-step 1. Shape depending on and a. The ellipse is stretched more along the -axis.The major axis is horizontal, length .The minor axis is vertical, length .Foci lie on the -axis at where .b. The ellipse is stretched more along the -axis.The major axis is vertical, length .The minor axis is horizontal, length .Foci lie on the -axis at where .c. The ellipse is symmetric in and ; it becomes a circle of radius (or ).Major and minor axes are the same length .2. Is a circle an ellipse? Why or why not?Yes — a circle is a special case of an ellipse. When the ellipse equation becomes\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}=1 \quad\Longrightarrow\quad x^{2}+y^{2}=a^{2},3. x- and y-intercepts for an ellipse centered at the originSet in the equation to get the x-intercepts:\frac{x^{2}}{a^{2}}=1 \Rightarrow x=\pm a,Set to get the y-intercepts:\frac{y^{2}}{b^{2}}=1 \Rightarrow y=\pm b,
