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Given we have the radius of this circle and the coordinates of A, how can I calculate the coordinate of C? C is also at the furthest part away from the line, or in other words a line going form A to C would be rotated 90 degrees from the line passing through A.

I tried using a substitution of the distance between point A and point C into the line equation for A, but I just can't seem to get the maths right for it.

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Not sure all the information you have, but assuming you have the vertex of the circle, example 3 gets the shortest distance, which would provide the line you need through the vertex as the other point of the line on the circle would be the longest distance. https://www.varsitytutors.com/hotmath/hotmath_help/topics/shortest-distance-between-a-point-and-a-circle

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