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tatekuhic

Guest

Circle Problems

Circle C1 has equation (x+2)^2 + (y+4)^2 = 64 and circle C2 has equation (x-h)^2 + (y-1)^2 = 81.
The distance between the center of the circles is 13.
1. Find all possible values of h
2. If a segment connecting the centers is drawn, let A be the intersection of the segment with C1 and B be the intersection of the segment with C2. Find AB.
3. Find the equation of the two circles that have the same center as C1 and are tangent with C2.

For part 1, I've already found out that h=-14, 10.  Could you please help me with the rest?  Thanks!  (Please include steps, not just the answers)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: Circle Problems

hi tatekuhic

Welcome to the forum.

If you have worked out h (well done!) then you know the full equation of both circles, and hence the centre of C2.

This means you can work out the equation of AB, and so you can substitute for y in each circle equation and so work out the coordinates of A and separately, B.  This will also give you the coordinates of the second point on AB where it crosses C2, lets call it point D.

Because AB gives the line that is a diameter of each circle, the tangents must go through A and D, so you have all you need to form the equations of these new circles, centre and a point on the circumference.

I haven't tried the question yet.  If you are still having trouble with any part, post again and I'll try to help some more.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

pamshaw

Member

Registered: 2021-12-07

Posts: 21

Re: Circle Problems

Hello,
The chord of a circle (mytutorsource.com/blog/chord-of-a-circle) is a line segment that links or unites two points on a circle's circumference.
According to the question,
C 1:  (x + 2) ^ 2 + (y + 4) ^ 2 = 64
C 2: (x - h) ^ 2 + (y - 1) ^ 2 = 81
Distance between the center of the circles = 13

Possible Values of h
You have already calculated that as h = -14 and h = 10
For Part 2,  If a segment connecting the centers is drawn, let A be the intersection of the segment with C1 and B be the intersection of the segment with C2. Find AB.
You may calculate the coordinates of A and B individually by substituting for y in each circular equation. This will also provide you with the coordinates for the second place on AB where it intersects C2. Since AB is the line that defines each circle's diameter, the tangents must pass through A and the second point, so you have everything you need to write the equations for these new circles' center and circumference point.