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Chapter 0: Problem 90

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 is \(x^2 + y^2 = 17\).

Step by step solution

01

Find the Centre of the Circle

The centre of the circle is halfway between the two end points of the diameter. The coordinates for the centre (\(a,b\)) can be calculated with the following formulas: \(a = (x_1 + x_2) / 2\) and \(b = (y_1 + y_2) / 2\). For the given points \((-4,-1)\) and \((4,1)\), we get \(a = (-4 + 4) / 2 = 0\) and \(b = (-1 + 1) / 2 = 0\). Therefore, the centre of the circle is at the origin (0,0).
02

Calculate the radius of the Circle

The radius \(r\) of the circle can be determined via the distance formula between two points: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). By applying this formula to one of the endpoints of the diameter and the centre of the circle, the radius becomes \(r = \sqrt{(4 - 0)^2 + (1 - 0)^2} = \sqrt{16 + 1} = \sqrt{17}\).
03

Write the Standard form of the Equation of the Circle

Substitute the values of \(a\), \(b\), and \(r\) into the standard form equation. The equation of the circle is thus \((x - 0)^2 + (y - 0)^2 = \(\sqrt{17}\)^2\), which simplifies to \(x^2 + y^2 = 17\).

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Most popular questions from this chapter

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)= \begin{cases}-x, & x \leq 0 \\ x^{2}-3 x, & x>0\end{cases} $$

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