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Question 1146376: Find the equation of the line that is tangent to the point of contact of the circles x2 + y2 + 2x - 4y -20 = 0 and x2 + y2 - 10x + 5y + 25 = 0.
Found 2 solutions by greenestamps, Alan3354:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The meaning of the question is clear; but it is not stated correctly.
A line can't be tangent to a point; it has to be tangent to a curve. The statement of the problem should say to find the line that is tangent to both circles at point of contact.
Completing the squares in both variables in both equations shows the first circle with center (-1,2) and radius 5 and the second with center (5,-2.5) and radius 2.5.
We could solve the pair of equations simultaneously to find the point of contact. However, the fact that the statement of the problem implies that the two circles are tangent to each other makes it easier to find the point of contact.
The horizontal and vertical distances between the two centers are 6 and -4.5; and the radius of the first circle is twice the radius of the second. So the point of contact is 2/3 of the distance from (-1,2) to (5,-2.5). 2/3 of the horizontal and vertical distances between the two centers puts us at (-1+4,2-3) = (3,-1).
The slope of the line containing the two centers is -3/4, so the slope of the line tangent to the two circles is 4/3.
An equation of the line with slope 4/3 passing through point (3,-1) is
Here is a graph: first circle red; second circle green; line containing the two center blue; tangent line purple.
Answer by Alan3354(69443)  (Show Source):
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