Question 11.4.3: Finding the Equation of a Hyperbola Find the standard form o...

Because the foci of the hyperbola (-5, 0) and (5, 0) are on the x-axis, the transverse axis lies on the x-axis. The center of the hyperbola is midway between the foci, at (0, 0). The standard form of such a hyperbola is

\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1

We need to find a² and b².

The distance a between the center (0, 0) to either vertex, (-4,0) \text { or }(4,0) , is 4; so a=4 \text { and } a^{2}=16. The distance c between the center (0, 0) to either focus, (-5, 0) or (5, 0), is 5; so c = 5 and c² = 25. Now use the equation b^{2}=c^{2}-a^{2} to find b^{2}=c^{2}-a^{2}=25-16=9 \text {. Substitute } a^{2}=16 \text { and } b^{2}=9 \text { in } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 to get the standard form of the equation of the hyperbola

\frac{x^{2}}{16}-\frac{y^{2}}{9}=1