Answer
Hyperbola.
Work Step by Step
${{x}^{2}}-\frac{{{y}^{2}}}{25}=1$
Therefore, both variables are squared, so the graph is not a parabola.
The subtraction sign indicates that the provided equation ${{x}^{2}}-\frac{{{y}^{2}}}{25}=1$ is a hyperbola.
We also know that:
$y=\frac{b}{a}x$ and $y=-\frac{b}{a}x$
where $a$ and $b$ are the $x\text{-intercept}$ and $y\text{-intercept}$ of the hyperbola, respectively.
Substitute $5$ for $a$ and $-5$ for $b$ in the formula:
$y=5x$ and $y=-5x$
Thus, the provided equation ${{x}^{2}}-\frac{{{y}^{2}}}{25}=1$ is a hyperbola with asymptotes $y=5x$ and $y=-5x$.
The intersecting points are calculated as follows:
Substitute $y=0$ in the provided equation
${{x}^{2}}=1$
Taking the square root on both sides:
$x=\pm 1$
Thus, the graph passes through the points $\left( 1,0 \right)$ and $\left( -1,0 \right)$.
Now graph the equation.
