The equation of an ellipse is 4 x squared plus y squared plus 10 y plus 13 equals 0.

Write the equation in standard form.

(1 point)
Responses

Start Fraction x squared over 4 End Fraction plus left-parenthesis y plus 5 right-parenthesis squared over 3 End Fraction equals 1
Image with alt text: Start Fraction x squared over 4 End Fraction plus left-parenthesis y plus 5 right-parenthesis squared over 3 End Fraction equals 1

Start Fraction x squared over 4 End Fraction plus Start Fraction left-parenthesis y plus 5 right-parenthesis squared over 12 End Fraction equals 1
Image with alt text: Start Fraction x squared over 4 End Fraction plus Start Fraction left-parenthesis y plus 5 right-parenthesis squared over 12 End Fraction equals 1

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
Image with alt text: 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

Start Fraction x squared over 12 End Fraction plus Start Fraction left-parenthesis y plus 5 right-parenthesis squared over 3 End Fraction equals 1

1 answer

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 \(y^2 + 10y\):

  1. Take half of 10, which is 5, and square it to get \(25\).
  2. 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.