To write the equation of the ellipse in standard form, we need to first rearrange the given equation and complete the square.
The given equation is:
\[ 4x^2 + y^2 + 10y + 13 = 0 \]
First, we can move 13 to the other side:
\[ 4x^2 + y^2 + 10y = -13 \]
Next, we complete the square for the \(y\) terms. We have \(y^2 + 10y\):
- Take half of 10, which is 5, and square it to get \(25\).
- Add and subtract \(25\) inside the equation:
\[ 4x^2 + (y^2 + 10y + 25 - 25) = -13 \] \[ 4x^2 + (y + 5)^2 - 25 = -13 \]
Now, rearranging gives:
\[ 4x^2 + (y + 5)^2 = -13 + 25 \] \[ 4x^2 + (y + 5)^2 = 12 \]
Now, we can divide everything by 12 to put it into standard form:
\[ \frac{4x^2}{12} + \frac{(y + 5)^2}{12} = 1 \]
This simplifies to:
\[ \frac{x^2}{3} + \frac{(y + 5)^2}{12} = 1 \]
Now, we can see that the standard form of the ellipse is:
\[ \frac{x^2}{3} + \frac{(y + 5)^2}{12} = 1 \]
Among the provided responses, the correct one is:
Start Fraction x squared over 3 End Fraction plus Start Fraction left-parenthesis y plus 5 right-parenthesis squared over 12 End Fraction equals 1.