Full Question:A circle has its center at (6,7) and goes through the point (1,4). A second circle is tangent to the first circle at the point (1,4) and has one-fourth the area. What are the coordinates for the center of the second circle?
I am having a lot of trouble with this question. I was able to answer an identical question, with the only difference being the second circle bore the same area. It was simple enough since that would mean they shared the same radius so I just applied the midpoint formula to the center of the other circle, the tangent point and the unknown center to find the coordinates.
The one-fourth area however complicated things and I'm been racking my brain trying to work through it.
By using the other problem (which bears the same point (besides the center of the second circle, obviously), I determined the area of the second circle which would be approximately 26.69. Dividing pi from that gives me 8.5 as the radius squared and approximately 2.92 as the actual area. Is this correct logic and if so, how would I then use this to find the center of the circle?
I need answers as soon as possible. Could you please explain how you obtained the answer. I actually don't learn anything if I just see what the correct numbers are.
for a circle, one quarter the area means half the radius, so the smaller circle has half the radius of the bigger one.
If two circles are tangent two each other then the point of tangency is on the line joining the centres of the circles
start at (6,7) you add the vector ( -5, -3) to get to (1, 4) then you either add or subtract half that vector to get to the centre of the small circle
$$C = (1,4) \pm (-2.5, -1.5) $$