Chapter 9.5, Problem 41PPS

To determine

To Find: The equation of the hyperbola

Answer to Problem 41PPS

  $\frac{x^2}{25}-\frac{y^2}{4}=1$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Centered at the origin}=\left(0,0\right)\\ \text{vertical traverse length =10}units\\ \text{Conjugate axis length }=4units\end{array}$

Formula Used:

  $\begin{array}{l}\frac{{\left(y-k\right)}^2}{a^2}-\frac{{\left(x-h\right)}^2}{b^2}=1\\ \frac{y^2}{a^2}-\frac{x^2}{b^2}=1\\ y=\frac{b}{a}xor=-\frac{b}{a}x\\ e=\frac{\sqrt{a^2+b^2}}{a}\\ l=\frac{2b^2}{a}\\ \left(h-a,k\right)\&\left(h+a,k\right)\\ \left(h,k-c\right)\&\left(h,k+c\right)\end{array}$

Calculation:

  $\begin{array}{l}\text{Centered at the origin}=\left(0,0\right)\\ \text{vertical traverse length =}10units\\ \text{Conjugate axis length }=14units\\ h=k=0\\ 2a=10\\ \Rightarrow a=5\\ 2b=4\\ \Rightarrow b=2\\ \frac{x^2}{25}-\frac{y^2}{4}=1\end{array}$

Conclusion:

Thus the equation of hyperbola is $\frac{x^2}{25}-\frac{y^2}{4}=1$