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Chapter 1: Problem 75

Write the standard form of the equation of the circle with the given characteristics. Endpoints of a diameter: \((0,0),(6,8)\)

Short Answer

Expert verified
The standard form of the equation for the circle with endpoints of a diameter at (0,0) and (6,8) is (x-3)^2 + (y-4)^2 = 25.

Step by step solution

01

Find the midpoint

The midpoint of the diameter can be calculated using the midpoint formula (d1 + d2) / 2. Let's denote the input endpoints as (x1,y1) and (x2,y2). So the midpoint, which is the center of the circle, has coordinates: cx = (x1 + x2) / 2 = (0 + 6) / 2 = 3 and cy = (y1 + y2) / 2 = (0 + 8) / 2 = 4
02

Calculate the Radius

The radius of the circle is the distance from the center to either endpoint. To get the radius, apply the distance formula r = sqrt[(x2-x1)^2 + (y2-y1)^2] / 2 . Now we'll plug in the center and one of the endpoints. Here, radius r = sqrt[(6-3)^2 + (8-4)^2] = sqrt[9 + 16] = sqrt[25] = 5
03

Write the Standard Form of the Circle's Equation

The standard form of a circle's equation is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius. Substituting the values for the center (3,4) and radius 5, the standard form equation for our circle becomes (x-3)^2 + (y-4)^2 = 5^2

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