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Chapter 1: Problem 76

Write the standard form of the equation of the circle with the given characteristics. Endpoints of a diameter: (-4,-1),(4,1)

Short Answer

Expert verified
The standard form of the equation of the circle with endpoints of a diameter at (-4,-1) and (4,1) is \(x^2 + y^2 = 17\).

Step by step solution

01

Find the Coordinates of the Center

The midpoint of a line segment with endpoints \((-4, -1)\) and \((4, 1)\) gives us the coordinates of the center of the circle. The formula for the midpoint of a line segment is \(( \frac{x_1+x_2}{2}, \frac{y_1+ y_2}{2})\). Substituting \(x_1= -4\), \(y_1= -1\), \(x_2= 4\), \(y_2= 1\) into the formula gives us: (\( \frac{-4+4}{2}, \frac{-1+ 1}{2})\) = \((0, 0)\), so the center of the circle is at the origin.
02

Find the Radius

The radius can be determined by finding the distance between the center of the circle and either endpoint of the diameter. The formula for distance is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substituting \(x_1= 0\), \(y_1= 0\), \(x_2= -4\), \(y_2= -1\) into the formula gives us: \(\sqrt{(-4 - 0)^2 + (-1 - 0)^2}\) = \(\sqrt{16 + 1} = \sqrt{17}\).
03

Write the Equation of the Circle in Standard Form

The standard form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) are the coordinates of the center, and \(r\) is the radius. Substituting \(h= 0\), \(k= 0\), and \(r = \sqrt{17}\) into the formula gives us: \((x - 0)^2 + (y - 0)^2 = (\sqrt{17})^2 = 17\), which simplifies to \(x^2 + y^2 = 17\)

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