Distance and Midpoint Formulas; Circles

The Distance Formula The distance, d, between the points (x1, y1) and (x2,y2) in the rectangular coordinate system is

Example Find the distance between (-1, 2) and (4, -3). Solution Letting (x1, y1) = (-1, 2) and (x2, y2) = (4, -3), we obtain

The Midpoint Formula Consider a line segment whose endpoints are (x1, y1) and (x2, y2). The coordinates of the segment's midpoint are To find the midpoint, take the average of the two x-coordinates and of the two y-coordinates.

Text Example Find the midpoint of the line segment with endpoints (1, -6) and (-8, -4). Solution To find the coordinates of the midpoint, we average the coordinates of the endpoints. (-7/2, -5) is midway between the points (1, -6) and (-8, -4).

Definition of a Circle A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The fixed distance from the circle’s center to any point on the circle is called the radius. Radius: r (x, y) (h, k) Center Any point on the circle

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

Example Find the center and radius of the circle whose equation is (x – 2)2 + (y + 3)2 = 9 and graph the equation. Solution In order to graph the circle, we need to know its center, (h, k), and its radius r. We can find the values of h, k, and r by comparing the given equation to the standard form of the equation of a circle. (x – 2)2 + (y + 3)2 = 9 h is 2. (x – 2)2 + (y – (-3))2 = 32 k is –3. r is 3.

Example cont. Find the center and radius of the circle whose equation is (x – 2)2 + (y + 3)2 = 9 and graph the equation. Solution 1 2 3 4 5 -2 -1 -3 -4 -5 -6 (2, -3) We see that h = 2, k = -3, and r = 3. Thus, the circle has center (2, -3) and a radius of 3 units. Plot the center, (2, -3), and find 3 additional points by going up, right, down, and left of the center by 3 units.

General Form of the Equation of a Circle The general form of the equation of a circle is x2 + y2 + Dx + Ey + F = 0. Complete the square:

Example-Completing the Square Given the equation: