Question

Use the given equation to answer the following questions.

16x^2-96x+4y^2=-80.

Find the vertices and foci of the ellipse.

How do you complete the square on this problem? Thanks.

Answer 1

\[16x^2 - 96x + 4y^2 = - 80\]\[16\left( x^2 - 6x \right) + 4y^2 = - 80\]we need to add +9 to complete x^2 - 6x, but we need to multiply with 16 so we need to add +144 to both sides\[16\left( x^2 - 6x \right) + 4y^2 + 144 = -80 + 144\]\[16\left( x^2 - 6x + 9 \right) + 4y^2 = 64\]\[16\left( x - 3 \right)^2 + 4y^2 = 64\]\[\frac{\left( x - 3 \right)^2}{4} + \frac{y^2}{16} = 1\]