Find the equation of a circle of radius 5 which is touching another circlex2 + y2 – 2x – 4y – 20 = 0 at (5, 5).

x² - 2x + y² - 4y – 20 = 0

x² - 2x + 1 +y² - 4y +4 – 20 – 5 = 0

(x – 1)² + (y – 2)² = 25

(x – 1)² + (y – 2)² = 5²

Since, the equation of a circle having centre (h,k), having radius as "r" units, is

(x – h)² + (y – k)² = r²

Centre = (1, 2)

Point of Intersection = (5, 5)

It intersects the line into 1: 1, as the radius of both the circles is 5 units.

Using Ratio Formula,

Ratio = m₁ : m₂

Assuming the co-ordinates of the centre of the circle be (p,q)

p + 1 = 10, q + 2 = 10

p = 9 & q = 8

Co-ordinates = (9,8)

Equation is,

(x – h)² + (y – k)² = r²

(x – 9)² + (y – 8)² = 5²

x² - 18x + 81 + y² - 16y + 64 = 25

x² - 18x + y² - 16y + 145 – 25 = 0

x² - 18x + y² - 16y + 120 = 0

Hence, the required equation is x² - 18x + y² - 16y + 120 = 0.