Minimum Distance between Circle and Line
Point P moves on the circle and point Q moves on the line Evaluate the minimum value of the length
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1 solution
The circle can be written as: (x + 1)^2 + (y - 2)^2 = 5. The given line: y = 2x - 6, has a line perpendicular to it with slope = (-1/2). We would like this perpendicular line to pass through the circle center whose coordinates are (-1,2). So if the equation of the perpendicular line is y = (-1/2)x + b, when x = -1, y = 2, so 2 =1/2 + b , and b = 3/2, so y = (-1/2)x + 3/2. Next, we seek the coordinates of the intersection of the circle and the perpendicular line. Substituting for y in the circle equation, (x + 1)^2 + ((-1/2)x - 1/2)^2 = 5, or (x + 3)(x - 1) = 0, so x = 1. Then y = (-1/2) + 3/2 = 1. So the coordinates of the intersection are (1,1). Since the slope is (-1/2), the line passes through (3,0). (We come down 1 and 2 units to the right). This point is on the given line, so the shortest distance is the distance from (1,1) to (3,0) = sqrt((3 -1)^2 + (1 -0)^2) = sqrt(5).. Ed Gray