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. 9x² +25y² - 36x + 50y – 164 = 0

Verified step by step guidance

1. Start with the given equation: $\mathrm{9x}{2}^{}+25y^2-36x+50y-164=0$.
2. Group the x terms and y terms together and move the constant to the other side: $9x^2\; -\; 36x\; +\; 25y^2\; +\; 50y\; =\; 164$.
3. Factor out the coefficients of the squared terms from the x and y groups: $9(x^2\; -\; 4x)\; +\; 25(y^2\; +\; 2y)\; =\; 164$.
4. Complete the square for each group inside the parentheses: For $x^2\; -\; 4x$, add and subtract ; for $y^2\; +\; 2y$, add and subtract .
5. Rewrite the equation including the completed squares and adjust the right side accordingly by adding $9\; imes\; 4$ and $25\; imes\; 1$ to 164, then express the left side as perfect square trinomials.

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 constants. This technique helps transform the given equation into a standard form, making it easier to identify the center and radii of the 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 for straightforward graphing and analysis of the ellipse's properties.

Foci of an Ellipse

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