Problem 89 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What happens to the shape of the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) as \(\frac{c}{a} \rightarrow \infty,\) where \(c^{2}=a^{2}+b^{2} ?\)

Step-by-Step Solution

Answer

The shape of the graph of \( \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) will approach to a pair of straight lines as \( \frac{c}{a} \rightarrow \infty \),where \( c^{2}=a^{2}+b^{2} \).

Step 1: Understand the role of parameters in the equation of the hyperbola

The general equation for a hyperbola centered at the origin is \( \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \). The denominators \( a^{2} \) and \( b^{2} \) represent the squares of the lengths of the semi-major and semi-minor axes respectively.

Step 2: Analyze the relationship between \( a \), \( b \), and \( c \)

We are given that \( c^{2}=a^{2}+b^{2} \). This is similar to the Pythagorean relation in an ellipse which implies that \( c \) is the distance from the center of the hyperbola to either focus.

Step 3: Determine the effect of \( \frac{c}{a} \rightarrow \infty \) on the shape of the graph

When \( \frac{c}{a} \) approaches infinity, it implies that \( c \) is much larger than \( a \). Since \( c \) is the distance to either focus, as it increases, the foci move farther away from the center. Therefore, as \( \frac{c}{a} \rightarrow \infty \), the hyperbola graph will more resemble a pair of straight lines.