Math 1314 Section 2.8 Notes 2.8 Distance and Midpoint Formulas-Circles

Distance Formula: Used to find the distance between two points  x1, y1 and x2, y2 

22 distance()()xxyy2121 Example 1: Find the distance between A(4,8) and B(1,12)

distance(14)(128) 22 distance(3)(4) 22 distance91625  5

Example 2: Find the distance between: a/ A(2, 7) and B(11, 9)

b/ C(-5, 8) and D(2, - 4)

Math 1314 Section 2.8 Notes

c/ M3 3, 5 and N 3, 4 5

Midpoint Formula: Used to find the center of a line segment Ax1, y1  and Bx2, y2 

x2 x 1 y 2 y 1 midpoint  , 22

Example 3: Find the midpoint between A(4,8) and B(1,12)

xxyy2121 midpoint,  22 1 4 12 8 midpoint  , 22 midpoint  5 ,10 2

Math 1314 Section 2.8 Notes

Example 4: Find the midpoint between: a/ A(2, 7) and B(14, 9)

b/ C(-5, 8) and D(2, - 4)

c/ M 18,  4 and N 2, 4

Math 1314 Section 2.8 Notes Circles Definition of Circle: A circle is a set of points in a plane that are located a fixed distance, called the radius from a given point in the plane called the center.

Standard Form of the Equation of a Circle: The standard form of the equation of a circle with the center (h, k) and the radius r is (x – h)2 + (y – k)2 = r2

Example 1: State the center and radius of each circle then graph it. a/ (x – 5)2 + (y + 3)2 = 4

b/ (x + 7)2 + (y – 2)2 = 1

Math 1314 Section 2.8 Notes Example 2: Write the equation of the circle in standard form given: a/ Center C(3, -5) and radius r = 7.

b/ Center C(0, 2) and passing through (3, -1).

Math 1314 Section 2.8 Notes General Form: The equation of a circle can be written as x2 + y2 + ax + by + c = 0, this is called general form. Remark: Equation x2 + y2 + ax + by + c = 0 does not always represent a circle. It may represent a point or no graph.

Example 2: Write the equation of the circle in standard form and general form given: Center (-3, 8)and radius is 6. Since the center is (-3, 8), we have h = - 3, k = 8 and the radius is 6, so r = 6. Then the equation is (x + 3)2 + (y – 8)2 = 36 (standard form) And expand the squares and simplify to obtain the general form x2 + y2 + 6x – 16y + 37 = 0.

Example 3: Write the equation of the circle in standard form and general form given: Center (2, -9) and radius is 11 .

Math 1314 Section 2.8 Notes Find Center and Radius of a Circle Given a General Equation We need to write the equation of a circle in a standard form. Example 4: Find center and radius of a circle whose equation is a/ x2  y2 10x  2y 17  0

Math 1314 Section 2.8 Notes b/ x2  y2  4x  8y  20  0

2 2 c/ x  y  4x  8y  25  0