
analytic geometry
:confused:
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.
THANK YOU IN ADVANCE!!!!!:)

Let circumference Q be of radius r. Its center, Q(x,y)
Circumference M has a radius of 10, and its center is M(0,0)
Circumference P has a radius of 2, and its center is P(8,0)
MQ = sqrt[(x0)^2 +(y0)^2] = 10 r
So, r = 10  sqrt[x^2 +y^2] (i)
PQ = sqrt[(x8)^2 +(y0)^2] = 2 +r
So, r = sqrt[x^2 16x +64 +y^2] 2 (ii)
r = r,
10  sqrt[x^2 +y^2] = sqrt[x^2 16x +64 +y^2] 2
12  sqrt[x^2 +y^2] = sqrt[x^2 16x +64 +y^2]
Square both sides,
144 24sqrt[x^2 +y^2] +x^2 +y^2 = x^2 16x +64 +y^2
24sqrt[x^2 +y^2] = 16x 80
Divide both sides by 8,
3sqrt[x^2 +y^2] = 2x +10
Square both sides,
9[x^2 +y^2] = 4x^2 +40x +100
9x^2 +9y^2 = 4x^2 +40x +100
5x^2 40x +9y^2 = 100
5[x^2 8x] +9y^2 = 100
5[x^2 8x +16 16] +9y^2 = 100
5[x^2 8x +16] +9y^2 = 100 +80
5[(x 4)^2] +9y^2 = 180 an ellipse.
Divide both sides by 180,
[(x 4)^2]/36 +(y^2)/20 = 1
[(x 4)^2]/(6^2) +[(k 0)^2]/(2sqrt(5))^2 = 1 ***
That is the the equation of the locus of the centers of the circles of the Q kind.
It is an ellipse whose center is at (4,0), whose semimajor axis is 6, and whose semiminor axis is 2sqrt(5).