Identify the type of the conic section represented by the equation \(9x^2 - 4y^2 = 36\).
Answer
Step 3: This equation is in the standard form of a hyperbola where \(a^2 = 4\) and \(b^2 = 9\). The terms on the left side of the equation are subtracted, which indicating that the conic section is a hyperbola.
Step 1 ：Step 1: Rewrite the equation in the standard form of conic sections. The standard forms are: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] for ellipse, \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\] for hyperbola, \[(x-h)^2 + (y-k)^2 = r^2\] for circle, and \[y = ax^2 + bx + c\] or \[x = ay^2 + by + c\] for parabola.
Step 2 ：Step 2: Divide the equation by 36 to get \[\frac{x^2}{4} - \frac{y^2}{9} = 1\].
Step 3 ：Step 3: This equation is in the standard form of a hyperbola where \(a^2 = 4\) and \(b^2 = 9\). The terms on the left side of the equation are subtracted, which indicating that the conic section is a hyperbola.
