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Chapter 4: Problem 22

Find two choices for \(b\) such that \((b, 4)\) is on the circle with radius 3 centered at (-1,6) .

Short Answer

Expert verified
The two choices for the coordinate \(b\) such that the points \((b, 4)\) lie on the circle with a radius of 3 centered at \((-1,6)\) are \(b = -1 + \sqrt{5}\) and \(b = -1 - \sqrt{5}\).

Step by step solution

01

Write the equation of the circle

Using the circle's center and radius, write the equation of the circle: \((x + 1)^2 + (y - 6)^2 = 3^2\)
02

Plug in the given values

Now plug in the y-coordinate (4) into the equation: \((x + 1)^2 + (4 - 6)^2 = 3^2\)
03

Simplify the equation

Simplify the equation by squaring the terms inside the parenthesis and combining like terms: \((x + 1)^2 + (-2)^2 = 9\) \((x + 1)^2 + 4 = 9\)
04

Solve for x (b)

Move all the terms to one side and solve the equation for \(x\), which represents our coordinate \(b\): \((x + 1)^2 = 5\) \(x + 1 = \pm \sqrt{5}\) \(x = -1 \pm \sqrt{5}\) So, we have two different choices for the coordinate \(b\): 1. x = -1 + 鈭�5 => b = -1 + 鈭�5 2. x = -1 - 鈭�5 => b = -1 - 鈭�5 Now, we found two choices for the coordinate \(b\) such that the points \((b, 4)\) lie on the circle with a radius of 3 centered at \((-1,6)\).

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