Chapter 9.5, Problem 22PPS

To determine

To Plot: The graph of given hyperbola

Explanation of Solution

Given:

  $25x^2-4y^2+48y-100x+144=0$

Formula Used:

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

Calculation:

  

  $\begin{array}{l}25x^2-4y^2+48y-100x+144=0\\ \Rightarrow -4{\left(y-6\right)}^2+25{\left(x-2\right)}^2=100\\ \Rightarrow \frac{{\left(x-2\right)}^2}{4}-\frac{{\left(y-6\right)}^2}{25}=1\\ a=2\\ b=5\\ \left(h,k\right)=\left(2,6\right)\\ p=\frac{b^2}{\sqrt{a^2+b^2}}=\frac{25}{\sqrt{25+4}}\\ =\frac{25}{\sqrt{29}}\\ \text{Asymptotes}\\ y=\pm \frac{2}{3}\left(x-2\right)+6\\ Vertices\\ \left(0,6\right)\&\left(4,6\right)\end{array}$