equation of a circle?

Question

given three circles. the first circle is C1 given the equation of C1 is X^2+Y^2-10X-4Y+28=0 and the second circle C2 given the equation of C2 is X^2+Y^2-16X+4Y+52=0. the centre of the three circles lie on a line. find the equation of the third circle C3.

the three circles touch with one another. C3 is the biggest circle C1 and C2 lie inside the circle different C1 is the smallest.

Answer 1

There is not enough information here. If the center of C3 is on the line through the centers of C1 and C2, it could be ANYWHERE on that line. It could be between C1 center and C2 center or a billion lightyears away. And besides a center, a circle needs a radius. No information about that in the question.

Standard form for C1 is (x - 5)² + (y - 2)² = 1
center at (5,2), radius 1

and for C2, (x - 8)² + (y + 2)² = 16
center at (8,-2), radius 4

Answer 2

There is not enough information to find the equation for the third circle. All we can find is the equation of the line along which its center lies.

To find the centers and radii of the first two circles, complete the squares.

Circle C1:
x² + y² - 10x - 4y + 28 = 0
(x² - 10x + 25) + (y² - 4y + 4) = 1
(x - 5)² + (y - 2)² = 1
Center (h,k) = (5,2)

Circle C2:
x² + y² - 16x + 4y + 52 = 0
(x² - 16x + 64) + (y² + 4y + 4) = 16
(x - 8)² + (y + 2)² = 16
Center(h,k) = (8,-2)

The centers of the two circles define a line.
The slope of the line is:
m = ∆y/∆x = (2 + 2)/(5 - 8) = 4/-3 = -4/3

The equation of the line is:
y + 2 = (-4/3)(x - 8) = (-4/3)x + 32/3
y = (-4/3)x + 26/3

The center of circle C3 lies along this line, but there is no information in the question to tell us where along the line it lies or what the radius of the circle is.

Answer 3

there is no "the" third circle. any circle on the line that connects the centers of the two given circles is ok.