Answer
Hyperbola
Work Step by Step
$\frac{{{x}^{2}}}{25}-\frac{{{y}^{2}}}{49}=1$
It is written as:
$\frac{{{\left( x-0 \right)}^{2}}}{{{\left( 5 \right)}^{2}}}-\frac{{{\left( y-0 \right)}^{2}}}{{{\left( 7 \right)}^{2}}}=1$
Now, compare it with the standard equation of a hyperbola:
$\frac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}-\frac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1$
Then we get;
$a=5,\text{ }b=7$
Thus, the vertices of the hyperbola are $\left( 0,\pm 7 \right)$.
Since,
$c=\sqrt{{{a}^{2}}+{{b}^{2}}}$
Then, we find:
$\begin{align}
& c=\pm \sqrt{{{a}^{2}}+{{b}^{2}}} \\
& =\pm \sqrt{{{\left( 5 \right)}^{2}}+{{\left( 7 \right)}^{2}}} \\
& =\pm \sqrt{25+49} \\
& =\pm \sqrt{74}
\end{align}$
Therefore, the foci of the hyperbola are $\left( 0,\pm \sqrt{74} \right)$.
Using this information, we graph.
