Objective: Write an equation of a Circle Equations of Circles Mr. Peter Richard Wears Cool Glasses Too! Objective: Write an equation of a Circle And do other stuff too!

THE DISTANCE AND MIDPOINT FORMULAS Investigating Distance: 1 Plot A(2,1) and B(6,4) on a coordinate plane. Then draw a right triangle that has AB as its hypotenuse. B (6, 4) A (2, 1) x y 2 Find and label the coordinates of the vertex C. 3 Find the lengths of the legs of ABC. AB 5 yB – yA 3 4 – 1 4 Use the Pythagorean theorem to find AB. Remember: a 2 + b 2 = c 2 xB – xA 6 – 2 4 C (6, 1) 4 2 + 3 2 = c 2 16 + 9 = c AB = 5 16 + 9 = c 2 25 = c

Finding the Distance Between Two Points The steps used in the investigation can be used to develop a general formula for the distance between two points A(x 1, y 1) and B(x 2, y 2). Using the Pythagorean theorem x 2 – x 1 y2 – y1 d x y C (x 2, y 1 ) B (x 2, y 2 ) A (x 1, y 1 ) a 2 + b 2 = c 2 You can write the equation (x 2 – x 1) 2 + ( y 2 – y 1) 2 = d 2 Solving this for d produces the distance formula. THE DISTANCE FORMULA The distance d between the points (x 1, y 1) and (x 2, y 2) is d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

Compare with Distance Formula Equation of a Circle The center of a circle is (h, k) The radius of a circle is given by r The equation of a circle in standard form is (x – h)2 + (y – k)2 = r2 r = (x – h1) 2 + ( y – k) 2 Compare with Distance Formula THE DISTANCE FORMULA The distance d between the points (x 1, y 1) and (x 2, y 2) is d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

Given Center and Radius Write The Equation of the Circle Center is (-6, 10) with radius of 4 (x – h)2 + (y – k)2 = r2 (x+6)2 + (y-10)2 = 16

Try some on your own Find the center of the circle given the equation (9,-12) (-9,12)

Example 2 Find the equation of a circle with center of (-3,7) and a radius of 8 (x-3)2+(y+7)2=64 (x-3)2+(y+7)2=8 (x+3)2+(y-7)2=64 (x+3)2+(y-7)2=8

(x – h)2 + (y – k)2 = r2 Writing an Equation of a Circle Write the standard form of the equation of the circle that passes through the point (2,1) and whose center is the origin. (h,k) = (0,0) (x – h)2 + (y – k)2 = r2 This equation shows the pattern for all circles Whose midpoint is at the origin.

Try This One Yourself! Find the equation of a circle with a center of (-4,5) and has the point (3, 6) (x-4)2+(y+5)2=50 (x-4)2+(y-5)2=49 (x+4)2+(y-5)2=48 (x+4)2+(y-5)2=50

r is the radius radius is 4 units Graphing an Equation of a Circle Standard Equation of a Circle (Center at Origin) r is the radius radius is 4 units

Graphing an Equation of a Circle Graph the equation. Give the radius of the circle.

Graphing an Equation of a Circle Graph the equation. Give the radius of the circle.

Notes Over 10.3 Graphing an Equation of a Circle Graph the equation. Give the radius of the circle.

Time to Circularize! Quiz: Page 564/565 # 20, 24, 28, 30, 50 Homework: # 19, 23, 27, 29, 49