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A Word Problem

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A circle has the equaton (x-2)^2+(y+3)^2=4 and a line has the equation 5y-4x+20=0.

a) What is the distance from the center of the circel to the line.

b) What is the greatest distance from a point on the circle to the line?

Thanks!

Jun 15, 2020

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a)
Center = (2, -3).

Using the point-line distance formula:

$$\quad \text{Distance between }(x_0, y_0)\text{ and the line }ax+by+c=0\\ = \dfrac{\left|ax_0 + by_0 + c\right|}{\sqrt{a^2 + b^2}}$$

Plugging in $$(x_0, y_0, a, b, c) = (2, -3, -4, 5, 20)$$ gives the answer.

b)

In this case, you plot a straight line perpendicular to 5y - 4x + 20 = 0, which passes through the center of the circle. This straight line intersects the circle twice. For each intersection, calculate the point-line distance between the intersection and the straight line 5y - 4x + 20 = 0. The smaller one is the minimum distance, and the larger one is the required answer.

Jun 15, 2020