Write ellipse in standard form 4x^2+25y^2-50y=75

Answer 1

To write the ellipse in standard form, we need to complete the square for the y-terms.

Starting with the equation 4x^2 + 25y^2 - 50y = 75, we can rewrite it as:

4x^2 + 25(y^2 - 2y) = 75

To complete the square for the y-terms, we take half of the coefficient of y (-2) and square it.

So, (-2/2)^2 = 1

Add and subtract this value inside the parentheses:

4x^2 + 25(y^2 - 2y + 1 - 1) = 75

4x^2 + 25((y-1)^2 - 1) = 75

Expand the terms inside the parentheses:

4x^2 + 25(y-1)^2 - 25 = 75

Now, we have the equation in standard form:

4x^2 + 25(y-1)^2 = 100