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. 25x²+4y² – 150x + 32y + 189 = 0

Verified step by step guidance

1. Start by rearranging the given equation: \(25x^2 + 4y^2 - 150x + 32y + 189 = 0\). Move the constant term to the other side: \(25x^2 - 150x + 4y^2 + 32y = -189\).
2. Complete the square for the \(x\) terms: Factor out the coefficient of \(x^2\), which is 25, from the \(x\) terms: \(25(x^2 - 6x)\).
3. To complete the square, take half of the coefficient of \(x\) (which is -6), square it, and add it inside the parentheses: \((x^2 - 6x + 9)\). Remember to balance the equation by adding \(25 \times 9\) to the other side.
4. Complete the square for the \(y\) terms: Factor out the coefficient of \(y^2\), which is 4, from the \(y\) terms: \(4(y^2 + 8y)\).
5. To complete the square, take half of the coefficient of \(y\) (which is 8), square it, and add it inside the parentheses: \((y^2 + 8y + 16)\). Remember to balance the equation by adding \(4 \times 16\) to the other side.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique involves rearranging the equation and adding or subtracting a constant to create a binomial squared. It is essential for converting equations into standard form, particularly for conic sections like ellipses.

Standard Form of an Ellipse

The standard form of an ellipse is given by the equation (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, respectively. Understanding this form is crucial for identifying the properties of the ellipse, including its orientation and dimensions.

Foci of an Ellipse

The foci of an ellipse are two fixed points located along the major axis, which are essential for defining the shape of the ellipse. The distance from the center to each focus is calculated using the formula c = √(a² - b²), where c is the distance to each focus, and a and b are the semi-major and semi-minor axes. Knowing the location of the foci helps in graphing the ellipse accurately.