Circles The Wheels on the Bus Go Round and Round (9.2)

POD Complete the square. x 2 + 6x – 7 = 0

POD Complete the square. x 2 + 6x – 7 = 0 x 2 + 6x = 7 x 2 + 6x + 9 = (x + 3)(x + 3) = 16 (x + 3) 2 = 4 2 What do you notice about this equation? What would its graph look like?

Today we start talking about conics Conics are also referred to as “quadratic relations.” The equations for these types of relations follow a certain kind of pattern. Ax 2 + Bxy + Cy 2 +Dx + Ey + F = 0 Some of the coefficients could be zero. Then what happens to that term?

Today we start talking about conics The first conic (or “conic section”) we’ll look at is circles. What is the definition for a circle? What are that point and that distance called?

Today we start talking about conics The first conic (or “conic section”) we’ll look at is circles. A circle: The set of points on a plane equidistant from a given point. What are that point and that distance called? The center and the radius.

Circle equations—on the origin If the center of the circle is on the origin, then the equation for the circle is given by x 2 + y 2 = r 2 where r is the radius. Which coefficients equal 0 in the general equation? Ax 2 + Bxy + Cy 2 +Dx + Ey + F = 0

Circle equations– on the origin If the center of the circle is on the origin, then the equation for the circle is given by x 2 + y 2 = r 2 where r is the radius. Sketch each of the following circles. 1. x 2 + y 2 = x 2 + y 2 = x 2 + y 2 = 3

Circle equations– on the origin Sketch each of the following circles. 1. x 2 + y 2 = x 2 + y 2 = x 2 + y 2 = 3

Circle equations– off the origin If the center of the circle is off the origin, so that it’s at the point (h, k), then the equation is given as (x – h) 2 + (y – k) 2 = r 2 Sketch each of the following circles. 1. (x – 2) 2 + (y – 3) 2 = 9 2. (x + 4) 2 + (y – 1) 2 = 9 3. x 2 + (y+2) 2 = 16 What do you need to know to graph them?

Circle equations– complete the square When the square is obvious in the equation, it’s easy to find the center and radius. Sometimes we have to get the equation into that squared form. That’s when we Complete the Square. And sometimes we have to complete it for both the x and y variables.

Circle equations– complete the square Complete the square and graph this circle. x 2 + y 2 + 6x – 12 = 0 The Method: 1. Move like terms in place and constants to the right hand side. 2. Complete the square(s). 3. Factor completed square.

Circle equations– complete the square 1. Move like terms in place and constants to the right hand side. x 2 + 6x + y 2 = Complete the square(s). x 2 + 6x y 2 = Factor completed square. (x + 3) 2 + y 2 = 21

Circle equations– complete the square Now sketch it. (x + 3) 2 + y 2 = 21

Circle equations– complete the square Do the same with these equations. x 2 + y 2 + 6x – 4y – 12 = 0 x 2 + y 2 – 10x +8y + 5 = 0 What terms from that general form for quadratic relations would be missing here?

Circle equations– complete the square Do the same with these equations. x 2 + y 2 + 6x – 4y – 12 = 0 (x + 3) 2 + (y – 2) 2 = 25 x 2 + y 2 – 10x +8y + 5 = 0 (x – 5) 2 + (y + 4) 2 = 36

Circle equations– complete the square Sketch them. (x + 3) 2 + (y – 2) 2 = 25 (x – 5) 2 + (y + 4) 2 = 36

Circle inequalities What happens when the equal sign changes to an inequality? Guess what these will look like. x 2 + y 2 ≤ 25 x 2 + y 2 ≥ 25

Circle inequalities What happens when the equal sign changes to an inequality? Guess what these will look like. x 2 + y 2 ≤ 25 x 2 + y 2 ≥ 25 One fills the inside and one fills the outside.

Finally Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2). What information do you need? How would you get it?

Finally Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2). What information do you need? The radius How would you get it? The distance formula

Finally Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2). The distance formula:

Finally Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2). The equation:

Finally Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2). Another student approach: