Complete the square to determine the type of curve represented by the equation. $4 x ^ { 2 } + 9 y ^ { 2 } - 16 x - 20 = 0$
A) $\frac { ( x - 2 ) ^ { 2 } } { 9 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$
, ellipse
B) $9 ( x - 2 ) ^ { 2 } - 4 y ^ { 2 } = 9$
, hyperbola
C) $\frac { x ^ { 2 } } { 9 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$
, ellipse
D) $\frac { ( x - 2 ) ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$
, hyperbola
E) $\frac { ( x - 2 ) ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$
, ellipse

Question

Quizplus · Accepted Answer

Completing the square for the $x$ terms gives $4(x^2-4x+4)$, which can be simplified to $4(x-2)^2$. Completing the square for the $y$ terms gives $9(y^2-4/3y)$, which can be simplified to $9(y-0)^2$. Moving the constant to the right side and dividing by 20 gives $\frac{(x-2)^2}{9}+\frac{y^2}{4}=1$. Therefore, the equation represents an ellipse. Choice E matches the completed square form and the correct curve.

Complete the Square to Determine the Type of Curve Represented 4x2+9y2−16x−20=04 x ^ { 2 } + 9 y ^ { 2 } - 16 x - 20 = 04x2+9y2−16x−20=0

Complete the Square to Determine the Type of Curve Represented $4 x ^ { 2 } + 9 y ^ { 2 } - 16 x - 20 = 0$