I'd start with these:
Then I'd think about this:
What's your plan for the rest?
The center and radius of circumference M are (0,0) and 10, respectively. The center and radius of circumference P are (8,0) and 2, respectively. Circumference Q is internally tangent to circumference M and externally tangent to circumference P.Of course that there are infinitely many circumferences Q which have this property. Determine and graph the equation of the curve that contains the centers of all the circumferences of the Q kind.
THANKS IN ADVANCE FOR YOUR HELP!!!!!
I've attached a drawing of the problem:
1. the blue circles are your circles P and M
2. the red circles are a few examples of circle Q
3. the black curve is the path of all centres of Q
By pure guessing I found out that this curve must be an ellipse with the equation:
I finally found a solution.
I've attached a drawing of the problem.
Let C(a, b) be the centre of the circle Q. The you have 2 right triangles (coloured grey and light blue). Use Pythagorean theorem:
1. Grey triangle:
2. Blue triangle:
I solved this system of equations for the variable r and plugged the result into the equation 1.:
. Now subtract the second equation from the first on:
. Solve for r:
. I plugged this value into the first equation:
1. Grey triangle: . Expand the bracket, rearrange the equation and multiply by . You'll get:
. Complete the square:
and divide by 36:
This is the the equation of an ellipse with the centre at (4, 0) and the axes: 6 and . Therefore my previous post is not correct: I said that the minor axis has a length of 4.5. Sorry.