Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 36x² +9y² - 216x = 0

Verified step by step guidance

1. Start with the given equation: \$36x^{2} + 9y^{2} - 216x = 0$.
2. Group the \(x\) terms and the \(y\) terms together: \$36x^{2} - 216x + 9y^{2} = 0$.
3. Factor out the coefficients of the squared terms to make completing the square easier: \$36(x^{2} - 6x) + 9y^{2} = 0$.
4. Complete the square for the \(x\) terms inside the parentheses. Take half of the coefficient of \(x\) (which is \(-6\)), square it, and add inside the parentheses, remembering to balance the equation by adding the equivalent value outside: half of \(-6\) is \(-3\), and \((-3)^{2} = 9\). So add and subtract \$9\( inside the parentheses: \)36(x^{2} - 6x + 9 - 9) + 9y^{2} = 0$.
5. Rewrite the completed square and simplify the constants: \$36(x - 3)^{2} - 36 imes 9 + 9y^{2} = 0$. Then move the constant term to the other side and divide the entire equation by the constant to get the ellipse in standard form.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² or (y - k)² by adding and subtracting appropriate constants. This technique helps transform the given equation into a recognizable conic section form, such as the standard form of an ellipse.

Standard Form of an Ellipse

The standard form of an ellipse equation is (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes. Converting to this form allows easy identification of the ellipse's size, shape, and position.

Foci of an Ellipse

The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their locations depend on the values of a, b, and the center, and are found using c² = |a² - b²|, where c is the distance from the center to each focus.