The equation of an ellipse is \( 4 x^{2}+y^{2}+10 y+13=0 \). Write the equation in standard form. (1 point) \( 0 \frac{x^{2}}{4}+\frac{(y+5)^{2}}{3}=1 \) \( 0 \frac{x^{2}}{3}+\frac{(y+5)^{2}}{12}=1 \) \( 0 \frac{x^{2}}{12}+\frac{(y+5)^{2}}{3}=1 \)

To rewrite the given equation of the ellipse \(4 x^{2} + y^{2} + 10 y + 13 = 0\) in standard form, we first rearrange and complete the square. Let's isolate the \(y\) part: 1. Rewrite: \(4x^{2} + y^{2} + 10y = -13\). 2. Complete the square for \(y\): \(y^{2} + 10y = (y + 5)^{2} - 25\). 3. Substitute back: \(4x^{2} + (y + 5)^{2} - 25 = -13\). 4. Rearrange to \(4x^{2} + (y + 5)^{2} = 12\). 5. Divide through by 12 to get the standard form: \(\frac{x^{2}}{3} + \frac{(y + 5)^{2}}{12} = 1\). So, the answer is: \( \frac{x^{2}}{3} + \frac{(y + 5)^{2}}{12} = 1 \). Now, onto some extra nuggets of knowledge! When working with ellipses, it’s delightful to know that these shapes aren’t just pretty; they have serious historical significance! The study of ellipses dates back to ancient civilizations. For instance, Apollonius of Perga, a Greek mathematician from the 3rd century BC, studied conic sections, including ellipses. He realized that they can be formed by the intersection of a cone and a plane! In the real world, ellipses have some fascinating applications; they pop up in physics, astronomy, and even engineering! For example, the orbits of planets around the sun are elliptical in shape. The sun is located at one of the foci of the ellipse, creating the beautiful dance of planetary motion. So, next time you look at the night sky, think of those stunning two-dimensional shapes that govern the universe!