HOW TO CHECK IF THE TWO CIRCLES DOES NOT INTERSECT

Consider two circles with radii r₁ and r₂

Let d be the distance between the centers of the two circles.

Problem 1 :

Circle P has center (−4, −1) and radius 2 units, circle Q has equation x² + y² − 2x + 6 y + 1 = 0. Show that the circles P and Q do not touch

Solution :

Center of the circle P = (-4, -1)

Center of the circle with Q :

x² + y² − 2x + 6 y + 1 = 0

x² − 2x + y² + 6 y + 1 = 0

(x - 1)² + (y + 3)² - 1² - 3² + 1 = 0

(x - 1)² + (y + 3)² = 9

Center of Q (1, -3) and radius = 3

Distance between P and Q = √(1 + 4)² + (-3 + 1)²

= √5² + (-2)²

= √(25 + 4)

Distance between two centers = √29

r₁ = 2, r₂ = 3

r₁ + r₂ = 2 + 3 ==> 5

√29 > 5

Distance between centers > sum of radii