In Exercises 51–60, 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: $\mathrm{36x}{x}^2+9y^2-216x=0$.
2. Group the x terms and y terms separately: $36x^2\; -\; 216x\; +\; 9y^2\; =\; 0$. Then, factor out the coefficients of the squared terms from the x and y groups: $36(x^2\; -\; 6x)\; +\; 9y^2\; =\; 0$.
3. Complete the square for the x terms inside the parentheses. Take half of the coefficient of x (which is -6), square it, and add and subtract that value inside the parentheses: half of -6 is -3, and (-3)^2 = 9. So, rewrite as $36(x^2\; -\; 6x\; +\; 9\; -\; 9)\; +\; 9y^2\; =\; 0$.
4. Rewrite the expression by grouping the perfect square trinomial and moving the constant terms to the other side: $36(x\; -\; 3)^2\; -\; 36\; imes\; 9\; +\; 9y^2\; =\; 0$, which simplifies to $36(x\; -\; 3)^2\; +\; 9y^2\; =\; 324$ after adding 36 × 9 to both sides.
5. Divide the entire equation by 324 to get the standard form of the ellipse: $\backslash frac\{(x\; -\; 3)^2\}\{9\}\; +\; \backslash frac\{y^2\}\{36\}\; =\; 1$. From here, identify the center, lengths of the axes, and use the formula for the foci of an ellipse to find their locations.

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 standard form, making it easier to analyze and graph conic sections like ellipses.

Standard Form of an Ellipse

The standard form of an ellipse equation is (x - h)^2/a^2 + (y - k)^2/b^2 = 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 identification of the ellipse's size, shape, and orientation.

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 are found using c^2 = |a^2 - b^2|, where c is the distance from the center to each focus along the major axis.