Questions: Without writing the equation (63 x^2-252 x+36 y^2-72 y=-18) in the standard form, state whether the graph of this equation is a parabola, circle, ellipse, or hyperbola. a. hyperbola c. circle b. parabola d. ellipse

Solution Steps

To determine the type of conic section represented by the equation \(63x^2 - 252x + 36y^2 - 72y = -18\), we need to analyze the coefficients of the squared terms. Specifically, we compare the coefficients of \(x^2\) and \(y^2\). If they are equal and have the same sign, it's a circle. If they have the same sign but are not equal, it's an ellipse. If they have opposite signs, it's a hyperbola. If only one variable is squared, it's a parabola.

The given equation is \(63x^2 - 252x + 36y^2 - 72y = -18\). We identify the coefficients of the squared terms:

• Coefficient of \(x^2\) is \(a = 63\).
• Coefficient of \(y^2\) is \(b = 36\).

To determine the type of conic section, we compare the coefficients \(a\) and \(b\):

• If \(a = b\), the conic is a circle.
• If \(a \neq b\) and \(a \cdot b > 0\), the conic is an ellipse.
• If \(a \cdot b < 0\), the conic is a hyperbola.
• If only one variable is squared, the conic is a parabola.

In this case, \(a = 63\) and \(b = 36\), and both are positive, so \(a \cdot b > 0\) and \(a \neq b\).

Final Answer