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# Positional relationship between two circles

Question:

The circle1 has the equation, x2 + y2 - 6x + 5 = 0 and the circle2 has the equation, x2 + y2 - 12y + 11 = 0. Which of the following statement is true?

1. Two circles intersect at two points.

2. Two circles is tangent at only one point.

3. Two circles have no intersection point.

Solution:

Find the center of the circle1.

x2 + y2 - 6x + 5 = 0
x2 - 6x + 32 - 32 + y2 + 5 = 0
(x - 3)2 + y2 = 32 - 5
(x - 3)2 + y2 = 9 - 5
(x - 3)2 + y2 = 4
(x - 3)2 + y2 = 22

The center of the circle1 is: (3, 0) and the radius of the circle1 is 2. Let the point A be the center of the circle1, then x-coordinate of the point A is 3 and y-coordinate of the point A is 0, and radius of the circle1 is r1 = 2

Now, we find the center of the circle2.

x2 + y2 - 12y + 11 = 0
x2 + y2 - 12y + 62 - 62 + 11 = 0
x2 + (y - 6)2 = 62 - 11
x2 + (y - 6)2 = 36 - 11
x2 + (y - 6)2 = 25
x2 + (y - 6)2 = 52

The center of the circle2 is: (0, 6) and the radius of the circle2 is 5. Let the point B is the center of the circle2, then the x-coordinate of the circle2 is 0 and y-coordinate of the center2 is 6, the radius of the center2 is: r2 = 5.

Now, we will find the distance between the center of the two circles. That is, the distance from the point A to the point B.

A (3, 0), r1 = 2, x1 = 3, y1 = 0
B (0, 6), r2 = 5, x2 = 0, y2 = 6
the distance from point A to point B,
|AB| = square root of [(x2 - x1)2 + (y2 - y1)2]
= square root of [(0 - 3)2 + (6 - 0)2]
= square root of (9 + 36)
= square root of 45
r1 + r2 = 2 + 5 = 7 = square root of 49
So, |AB| < r1 + r2

Because the distance from the center of the circle1 to the center of the circle2 is less than the sum of the radius of the circle1 and the radius of the circle2, so the circle1 intersects the circle2. So, the statement1 is correct. Watch the video for more details.

Note: If the distance from the center of the circle1 to the center of the circle 2 is equal to the sum of the radius of the circle1 and the radius of the circle2, then circle1 and circle2 are tangent to each other.

If the distance from the center of the circle1 to the center of the circle 2 is larger than the sum of the radius of the circle1 and the radius of the circle2, then circle1 and circle2 do not have intersection point.