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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 the given characteristics is \(x^{2} + y^{2} = 8\).

Step by step solution

01

Find the Center of the Circle

The center of the circle is simply the midpoint of the endpoints of the diameter. It can be found by taking the average of the x-coordinates and the y-coordinates separately. Therefore, the center (h,k) = \(\left(\frac{-4+4}{2},\frac{-1+1}{2}\right) = (0,0)\).
02

Find the Radius of the Circle

The radius of the circle is half of the distance between the two endpoints of the diameter. It can be calculated using the distance formula : \(r = \frac{\sqrt{(x_2-x_1)^{2} + (y_2-y_1)^{2}}}{2}\). Therefore, r = \(\frac{\sqrt{(4-(-4))^2 + (1-(-1))^2}}{2}\) = \sqrt{8}.
03

Write the Equation of the Circle

Substitute the center (h,k) and radius r into the standard form of a circle equation \((x-h)^2+(y-k)^2=r^2\). This gives us the final equation for the circle: \(x^{2} + y^{2} = 8\).

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Most popular questions from this chapter

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