I need this problem to be solved

Question

Circle C1 has centre (-2,5) and radius 7 . Circle C2 has centre (12,5)

a) Given that the two circles are tangent to each other, find the two possible values for

the radius of C2

b) Given instead that the radius of C2 is 16 , find the coordinates of the intersection

points of C1 and C2

Answer 1

center (-2,5) with radius 7 is Circle 1

(x+2)^2 + (y-5)^2 = 7^2

Circle 2 has center (12,5) which is horizontally to the right of Circle 1. the farthest right edge point of Circle 1 is (5,5)

for the farthest left edge point of Circle to be tangent to Circle 1 at (5,5), it would have the same point (5,5)

Circle 2 also has radius =7

(x-12)^2 + (x-5)^2 = 7^2

the two circles intersect at their point of tangency (5,5)

(5-12)^2 +(5-5)^2 = 7^2

(5+2)^2 + (5-5)^2 = 7^2

but a 2nd point of possible tangency is when both circles go through the point (-9,5), then Circle 2 has radius = 12--9=21

Circle 2 has radius 7 or 21

IF Circle 2 has radius = 16

(x-12)^2 + (y-5)^2 = 16^2 is the equation of Circle 2

(x+2)^2 + (y-5)^2 = 7^2 is the equation for Circle 1

subtract to eliminate the y variable

(x-12)^2 - (x+2)^2 = 16^2 -7^2

x^2 -24x +144 -(x^2 +4x +4) = 256-49 = 207

-24x -4x +140 = 207

-28x = 67

x = -67/28= about -2.39

(-67/28+2)^2 + y^2 -10y +25 = 49

(-11/28)^2 + y^2 -10y = 24

y^2 -10y = 24 -121/784 = (18816-121)/784 = 18,695/784

y-10+25 =18695/784 +25= (18,695+19,600)/784 =38,295/784

y-5 = sqr(38,295/784)

y = about 5 +/-6.99

y = -1.99 or 11.99

intersection points are about (-2.39, -1.99) and (-2.39, 11.99)

or about (-2.4, -2) and (-2.4, 12)