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Positional relationship between a circle and a point

Question:

A point P (5, 7) and a circle O: x2 + y2 - 8x - 6y = 0 lie in the xy-plane. Which of the following statement is true?

1. Point P lies inside of the circle.

2. Point P lies outside of the circle.

3. Point P lies on the circle.

Solution:

First, we need to find the center of the circle and the radius of the circle. The circle equation is: x2 + y2 - 8x - 6y = 0, we apply the completing square formula,

x2 - 8x + 42 - 42 + y2 - 6y + 32 - 32 = 0
(x - 4)2 + (y - 3)2 = 42 + 32
(x - 4)2 + (y - 3)2 = 16 + 9
(x - 4)2 + (y - 3)2 = 25
(x - 4)2 + (y - 3)2 = 52

Now, we get the center of the circle: O (4, 3) and the radius of the circle: r = 5. Point O is the center of the circle, its x-coordinate is 4 and y-coordinate is 3.

Now we need to find the distance between the center of the circle to the point P. The coordinate of the center of the circle is: O (4, 3) and the coordinate of the point P is: (5, 7). The distance from the center of the circle to the point P is |OP|.

O (4, 3), x1 = 4 and y1 = 3
A (5, 7), x2 = 5 and y2 = 7
|OP| = square root of [(x2 - x1)2 + (y2 - y1)2]
= square root of [(5 - 4)2 + (7 - 3)2]
= square root of (12 + 42)
= square root of 17
r = 5 = square root of 25
So, |OP| < r

Because the distance from the center of the circle to the point P is less than the radius of the circle, so the statement1 is correct. Watch the video for more details.

Note: If the distance from the center of the circle to the point P is equal to the radius of the circle, then the point P is on the circle. If the distance from the center of the circle to the point P is larger than the radius, then the point P is outside of the circle.

Is a point inside a circle, outside a circle or on a circle?

Look the figure above, because the distance from the center of the circle to the point P is less than the radius of the circle, so the point P is inside the circle.