Warm-Up Find the distance and the midpoint. 1. (0, 3) and (3, 4)

Graph and Write Equations of Circles Section 9-3 Graph and Write Equations of Circles

Vocabulary Circle – The set of all points (x, y) in a plane that are equidistant from a fixed point. Center – The fixed point of a circle. Radius – The distance r between the center and any point (x, y) on the circle.

Standard Equation of a Circle with Center at the Origin x2 + y2 = r2

Example 1 x2 + y2 = 4 Center : (0, 0) Radius: √4 = 2 Graph x2 – 4 = – y2. Identify the radius of the circle. Step 1: Rewrite the equation in standard form. x2 + y2 = 4 Center : (0, 0) Radius: √4 = 2 Step 2: Identify the center and radius. Step 3: Draw the circle.

Example 2 d = √(x2 – x1)2 + (y2 – y1)2 d = √(-4 – 0)2 + (7 – 0)2 The point (-4, 7) lies on a circle whose center is the origin. Write the standard form of the equation of the circle. Step 1: Use the distance formula. Because the point (-4, 7) lies on the circle, the circle’s radius r must be the distance between the center (o, 0) and (-4, 7). Step 2: Write the equation of the circle. d = √(x2 – x1)2 + (y2 – y1)2 d = √(-4 – 0)2 + (7 – 0)2 d = √(-4)2 + (7)2 x2 + y2 = 65 d = √ 65

Example 3 m = -6 – 0 5 – 0 m = 5 6 = -6 5 y – y1 = m(x – x1) What is an equation of the line tangent to the circle x2 + y2 = 61 at (5, -6)? Step 1: Find the slope of the line. A line tangent to a circle is perpendicular to the radius at the point of tangency. Step 2: Find the slope of the tangent line. m = -6 – 0 5 – 0 m = 5 6 = -6 5 Step 3: Use point-slope form to find the equation. y – y1 = m(x – x1) y – (-6) = 5(x – 5) 6 y = 5x – 61 6 6 y + 6 = 5x – 25 6 6

Homework Section 9-3 Pages 629 –630 1, 3 – 8, 10 – 12, 17, 18, 22, 23, 26, 28, 31, 32, 34, 44, 45, 46, 55, 58, 62