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Chapter 1: Problem 73

Write the standard form of the equation of the circle with the given characteristics. Center: \((-1,2) ;\) Solution point: \((0,0)\)

Short Answer

Expert verified
Therefore, the standard form of the equation of the circle with the given characteristics is \( (x+1)^2 + (y-2)^2 = 5 \)

Step by step solution

01

Calculate the Radius

Use the distance formula to calculate the radius. The distance r between any two points (x1, y1) and (x2, y2) is given by: \[ r = \sqrt{(x2-x1)^2 + (y2-y1)^2} \] Here, the center is at (-1, 2) and the solution point is at (0, 0), thus radius would be: \[ r = \sqrt{(0 - (-1))^2 + (0 - 2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
02

Substitute the Values into the Equation

Use the standard form of a circle, which is \( (x-h)^2 + (y-k)^2 = r^2 \) and substitute the values of h = -1, k = 2, and r = \sqrt{5}. Thus, \[ (x-(-1))^2 + (y-2)^2 = (\sqrt{5})^2 \] which simplifies to: \[ (x+1)^2 + (y-2)^2 = 5 \]

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