A circle with a radius 4 touches the line 6x-8y=1. Find the locus of its center.
The circle with radius equal to 4 touches the line 6x - 8y = 1. This implies that the center of the circle lies at a distance of 4 from the line.
Taking a point (x,y) on the locus of the center, we use the relation for determining the distance d of a point (x1, y1) from the line ax+by +c = 0, which is:
d = |ax1+by1+c|/ sqrt (a^2+b^2)
4 = |6x -8y - 1|/sqrt ( 36 + 64)
=> 4 = |6x -8y - 1|/sqrt ( 100)
=> 4*10 = 6x - 8y - 1 and -40 = 6x - 8y - 1
=> 6x - 8y - 41 = 0 and
= > 6x - 8y + 39 = 0
Therefore the locus of the center are the lines:
6x - 8y - 41 = 0 and 6x - 8y + 39 = 0
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