Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola.
A
A circle
B
An ellipse
C
A hyperbola
D
A parabola
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1
Start by writing down the given equation: \(4x^{2} - 9y^{2} = 36\).
Rewrite the equation in standard form by dividing every term by 36 to isolate 1 on the right side: \(\frac{4x^{2}}{36} - \frac{9y^{2}}{36} = \frac{36}{36}\).
Simplify the fractions: \(\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1\).
Recognize the form of the equation: it matches the standard form of a hyperbola, which is \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) or \(\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1\).
Conclude that since the equation has a subtraction between the squared terms and equals 1, the graph represents a hyperbola.