Position of a Point with Respect to a Circle

The position of a point with respect to a circle can be determined by comparing the distance from the point to the center of the circle with the radius of the circle. There are three possible positions for a point relative to a circle:

1. Inside the circle
2. On the circle
3. Outside the circle

Mathematical Representation

Let's consider a circle with center at point ( C(h, k) ) and radius ( r ). The general equation of the circle in the Cartesian coordinate system is:

[ (x - h)^2 + (y - k)^2 = r^2 ]

Now, let ( P(x_1, y_1) ) be a point whose position with respect to the circle we want to determine. We calculate the distance ( d ) from ( P ) to the center ( C ) using the distance formula:

[ d = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} ]

We compare ( d ) with the radius ( r ) to determine the position of ( P ):

• If ( d < r ), the point ( P ) lies inside the circle.
• If ( d = r ), the point ( P ) lies on the circle.
• If ( d > r ), the point ( P ) lies outside the circle.

Table of Differences and Important Points

Position	Condition	Formula Used	Example Point ( P(x_1, y_1) )	Circle Equation	Conclusion
Inside	( d < r )	( \sqrt{(x_1 - h)^2 + (y_1 - k)^2} < r )	( P(1, 2) )	( (x - 3)^2 + (y - 3)^2 = 4 )	( P ) is inside the circle
On the circle	( d = r )	( \sqrt{(x_1 - h)^2 + (y_1 - k)^2} = r )	( P(3, 4) )	( (x - 3)^2 + (y - 3)^2 = 1 )	( P ) is on the circle
Outside	( d > r )	( \sqrt{(x_1 - h)^2 + (y_1 - k)^2} > r )	( P(6, 6) )	( (x - 3)^2 + (y - 3)^2 = 4 )	( P ) is outside the circle

Examples to Explain Important Points

Example 1: Point Inside the Circle

Consider a circle with center ( C(3, 3) ) and radius ( r = 2 ). Let's find the position of point ( P(1, 2) ).

[ d = \sqrt{(1 - 3)^2 + (2 - 3)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} ]

Since ( \sqrt{5} < 2 ), point ( P ) lies inside the circle.

Example 2: Point On the Circle

Consider a circle with center ( C(3, 3) ) and radius ( r = 1 ). Let's find the position of point ( P(3, 4) ).

[ d = \sqrt{(3 - 3)^2 + (4 - 3)^2} = \sqrt{0 + 1} = 1 ]

Since ( d = 1 ), which is equal to the radius of the circle, point ( P ) lies on the circle.

Example 3: Point Outside the Circle

Consider a circle with center ( C(3, 3) ) and radius ( r = 2 ). Let's find the position of point ( P(6, 6) ).

[ d = \sqrt{(6 - 3)^2 + (6 - 3)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} ]

Since ( \sqrt{18} > 2 ), point ( P ) lies outside the circle.

Conclusion

The position of a point with respect to a circle is an important concept in geometry, particularly in problems involving tangents, secants, and chord properties. By understanding and applying the distance formula in relation to the radius of the circle, one can easily determine whether a point is inside, on, or outside a given circle.