Identify the conic by writing the equation in standard form. $4 x ^ { 2 } + 9 y ^ { 2 } + 40 x + 54 y + 145 = 0$
A) $\frac { ( x + 5 ) ^ { 2 } } { 9 } + \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$ ;ellipse
B) $( 3 x + 5 ) ^ { 2 } + ( 2 y + 1 ) ^ { 2 } = 167$ ;circle
C) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 422 } { 9 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 211 } { 2 } } = 1$ ;ellipse
D) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 73 } { 3 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 219 } { 4 } } = 1$ ;ellipse
E) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 167 } { 9 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 167 } { 4 } } = 1$ ;ellipse

Question

Identify the conic by writing the equation in standard form.​ $4 x ^ { 2 } + 9 y ^ { 2 } + 40 x + 54 y + 145 = 0$ ​
A) $\frac { ( x + 5 ) ^ { 2 } } { 9 } + \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$ ;ellipse
B) $( 3 x + 5 ) ^ { 2 } + ( 2 y + 1 ) ^ { 2 } = 167$ ;circle
C) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 422 } { 9 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 211 } { 2 } } = 1$ ;ellipse
D) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 73 } { 3 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 219 } { 4 } } = 1$ ;ellipse
E) $\frac { ( x + 5 ) ^ { 2 } } { \frac { 167 } { 9 } } + \frac { ( y + 3 ) ^ { 2 } } { \frac { 167 } { 4 } } = 1$ ;ellipse

Quizplus · Accepted Answer

To identify the conic and write the equation in standard form, complete the square for both $x$ and $y$. The given equation is $4x^2 + 9y^2 + 40x + 54y + 145 = 0$. Grouping $x$ and $y$ terms and completing the square:- For $x$: $4(x^2 + 10x) = 4[(x + 5)^2 - 25]$- For $y$: $9(y^2 + 6y) = 9[(y + 3)^2 - 9]$Substituting these into the original equation and simplifying gives:$$4[(x + 5)^2 - 25] + 9[(y + 3)^2 - 9] + 145 = 0$$$$4(x + 5)^2 + 9(y + 3)^2 - 100 - 81 + 145 = 0$$$$4(x + 5)^2 + 9(y + 3)^2 = 36$$Dividing through by 36 to get the equation in standard form for an ellipse:$$\frac{(x + 5)^2}{9} + \frac{(y + 3)^2}{4} = 1$$This matches choice (A), indicating the conic is an ellipse.

Identify the Conic by Writing the Equation in Standard Form 4x2+9y2+40x+54y+145=04 x ^ { 2 } + 9 y ^ { 2 } + 40 x + 54 y + 145 = 04x2+9y2+40x+54y+145=0

Identify the Conic by Writing the Equation in Standard Form $4 x ^ { 2 } + 9 y ^ { 2 } + 40 x + 54 y + 145 = 0$