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Chapter 0: Problem 89

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 of the circle with diameter endpoints (0,0) and (6,8) is \((x-3)^2 + (y-4)^2 = 25\).

Step by step solution

01

Find the Center of the Circle

The center of the circle is the midpoint of the segment having the endpoints of the diameter. Given the endpoints \((0,0)\) and \((6,8)\), the midpoint is calculated using the midpoint formula: \((x_1 + x_2) /2 , (y_1 + y_2) / 2\). Substituting the values, we get \((0+6)/2 = 3\) for the x-coordinate and \((0+8)/2 = 4\) for the y-coordinate. Therefore, the center of the circle is \((3, 4)\).
02

Calculate the Radius of the Circle

The radius of the circle is the distance between the center and any point on the circle. In this case, use one of the given endpoints of the diameter, for instance \((6,8)\). Use the distance formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to calculate it. Substituting the center point \((3,4)\) and the given point, we have radius \(r = \sqrt{(6-3)^2 + (8-4)^2} = 5\).
03

Write the Standard Form of the Equation of the Circle

Substituting the obtained center coordinates \((3, 4)\) and radius \(5\) into the general equation, we get \((x-3)^2 + (y-4)^2 = 5^2\). And, simplifying this gives the standard form of the equation of the circle: \((x-3)^2 + (y-4)^2 = 25\).

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