Question

homework 10: ellipse and hyperbola
score: 6/20 answered: 6/20
question 7
identify a and b for the ellipse with equation $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$.
$a =$
$b =$
question help: video 1 video 2

Explanation:

Step1: Find the value of \(a\)

The standard form of an ellipse is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\). For the \(x -\)intercepts, when \(y = 0\), we have \(\frac{x^{2}}{a^{2}}=1\), so \(x=\pm a\). From the graph, the \(x -\)intercepts are at \(x=-4\) and \(x = 4\). So \(a = 4\) (since \(a>0\)).

Step2: Find the value of \(b\)

For the \(y -\)intercepts, when \(x = 0\), we have \(\frac{y^{2}}{b^{2}}=1\), so \(y=\pm b\). From the graph, the \(y -\)intercepts are at \(y=-7\) and \(y = 7\). So \(b = 7\) (since \(b>0\)).

Answer:

\(a = 4\)
\(b = 7\)