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.

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.