identity the type of conic section that has the equation 9x^2 + 16y^2 = 144 and identify it's domain and range
The equation 9x^2 + 16y^2 = 144 represents an ellipse.
The standard form of an ellipse is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where (h,k) is the center of the ellipse, and a and b are the length of the major and minor axes, respectively.
Comparing the given equation with the standard form, we have:
\(a^2 = 16\), so \(a = 4\)
\(b^2 = 9\), so \(b = 3\)
The center of the ellipse is (0,0), and the major axis is along the x-axis while the minor axis is along the y-axis.
Therefore, the domain is \([-4, 4]\) and the range is \([-3, 3]\).