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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 of the circle with the given endpoints of a diameter is \((x-3)^2 + (y-4)^2 = (2.5)^2\).

Step by step solution

01

Calculate the center of the circle

The center of a circle represented by endpoints of a diameter can be easily calculated using the midpoint formula. The midpoint M of two points, (x1, y1) and (x2, y2), is calculated as \(M = [(x1+x2)/2 , (y1+y2)/2]\). Applying this formula to the given points, (0,0) and (6,8), the center (h, k) of the circle can be determined as \(h = (0+6)/2 = 3\) and \(k = (0+8)/2 = 4\), hence the center is (3, 4).
02

Calculate the radius of the circle

We have the coordinates of the center of the circle, so we need to find the radius of the circle. The radius, r is half of the diameter. The diameter's length is the distance between the center and any of the two points given, let鈥檚 choose (0,0). This can be found using the distance formula which is \(d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\). This simplifies to \(d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\). Therefore, the radius r is half of the diameter = 5/2 = 2.5.
03

Write the standard form of the equation

Now that we have the center (h, k)= (3, 4) and the radius r = 2.5, we can write the equation of the circle in its standard form. The standard form of the equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\). Inserting the values of h, k and r in the equation, the equation of the circle is given as \((x-3)^2 + (y-4)^2 = (2.5)^2\)

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