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Chapter 0: Problem 82

Tangent Line Find an equation of the line tangent to the circle \((x-1)^{2}+(y-1)^{2}=25\) at the point \((4,-3)\)

Short Answer

Expert verified
The equation of the tangent line to the circle \((x-1)^{2}+(y-1)^{2}=25\) at the point \((4,-3)\) is \(y = 0.75x -3\).

Step by step solution

01

Determine the Center and Radius of the Circle

From the equation \((x-1)^{2}+(y-1)^{2}=25\), the center of the circle can be determined to be (1, 1) and the radius \(r\) to be \(5\).
02

Calculate the Slope of the Radius

The radius extends from the center of the circle to any point on the circle. Consequently, it extends from (1, 1) to (4, -3). Applying the formula for the slope between two points: \(m = \frac{(y_{2} - y_{1})}{(x_{2} - x_{1})}\), we find \(m = \frac{(-3 - 1)}{(4 - 1)} = -1.33\).
03

Determine the Slope of the Tangent Line

The slope of the tangent line is the negative reciprocal of the slope of the radius. Consequently, the slope of the tangent is \(-1/m = -1/(-1.33) = 0.75\).
04

Write the Equation of the Tangent Line

We have enough information to write the equation of the tangent line. We know the slope (0.75) and a point on the line ((4, -3)). We plug these values into the point-slope form of a linear equation \(y - y_{1} = m(x - x_{1})\). We get \(y - (-3) = 0.75(x - 4)\), which when simplified gives \(y = 0.75x -3\).

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