Download 8.4—Equation of a Circle: Completing the Square

8.4—Equation of a Circle: Completing the Square
8.4 Warm Up
Match each perfect square trinomial to the equivalent expression written as the
square of a binomial. (Note: not all binomials will be used.)
8.4—Equation of a Circle: Completing the Square
Essential Question:
Suppose you are given an equation of a circle in general
form. How could you determine the center and radius of the
circle?
In order to rewrite the general form of an equation of a
circle in standard form, you must write the expanded
algebraic expressions as the sum of two perfect squares.
Standard Equation of a Circle with radius r and center (h, k):
(x – h)2 + (y – k)2 = r 2
If the center is the origin, then the standard equation is
x2 + y2 = r 2
General Equation of a Circle with radius r and center (h, k):
𝑥 2 + 𝑦 2 + 𝐶𝑥 + 𝐷𝑦 + 𝐸 = 0
Examples:
1. Complete the square to rewrite the quadratic equation below as a perfect square.
𝑥 2 + 6𝑥 + 11 = 5
(Show all algebraic steps and draw a diagram.)
2.
This is the original equation.
x2 + 6x – 7 = 0
Move the loose number over to the other side.
x2 + 6x
=7
**Rewrite the equation with a “+ space “ after
the x-term**
Take half of the x-term (that is, divide it by two)
(and don't forget the sign!), and square it. Add this
square to both sides of the equation.
Convert the left-hand side to squared
form. Simplify the right-hand side.
(x + 3)2 = 16
3.
This is the original equation.
Move the loose numbers over to the other side.
**Rewrite the equation, grouping x’s and y’s
together, with a “+ (Space) “ after the x-terms and a
“+ (Space) “ after the y-term.**
Take half of the x-term (that is, divide it by two)
(and don't forget the sign!), and square it. Add
this square to both sides of the equation.
Take half of the y-term (that is, divide it by two)
(and don't forget the sign!), and square it. Add
this square to both sides of the equation.
Convert the left-hand side to squared
form. Simplify the right-hand side.
𝑥 2 + 𝑦 2 + 6𝑥 + 4𝑦 − 3 = 0
Rewrite each equation in standard form. Identify the center and radius of each.
4. 𝑥 2 + 𝑦 2 − 20𝑥 − 12𝑦 + 111 = 0
5.
𝑥 2 + 𝑦 2 + 2𝑥 − 2𝑦 − 2 = 0
6.
𝑥 2 + 𝑦 2 − 8𝑦 = 0