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how to determine a point inside a circle or outside a circle

Question:

A circle has the equation x2 + y2 - 2y - 8 = 0. Point P has the coordinate (1, 3). Which of the following statement is true?

A. Point P is inside the circle.

B. Point P is on the circle.

C. Point P is outside the circle.

Solution:

First, we need to find the center of the circle and radius of the circle.

x2 + y2 - 2y - 8 = 0
x2 + y2 - 2y + 12 - 12 - 8 = 0
x2 + y2 - 2y + 12 - 9 = 0
x2 + y2 - 2y + 12 = 9
x2 + (y - 1)2 = 32

Therefore, the center of the circle is (0, 1) and the radius r = 3. Let Q is the center of the circle, so, the center coordinate of the circle is Q (0, 1).

Now, we will find the distance from the center of the circle to the point P. Let d is the distance from the center of the circle to the point P. Let Q is the first point and P is the second point. So, x1 = 0, y1 = 1, x2 = 1, y2 = 3.

d = |QP| = square root of (x2 - x1)2 + (y2 - y1)2
= square root of (1 - 0)2 + (3 - 1)2
= square root of (1 + 4)
= square root of 5
= 2.24

Because d < r, that is, the distance from the center of the circle to the point P is less than the radius of the circle, so the point P lies inside the circle. Watch the video for more details.

find a point inside a circle or outside a circle.