Position of a Point to a Circle

Testing Whether a Point Lies Within a Circle

Given a circle and a point, exactly one of three cases applies: the point lies inside the circle, on its circumference, or outside it.

The same comparison appears in simplified coverage models, such as deciding whether a location lies inside an idealized circular signal or radar range. The circle equation and point coordinates then classify the location exactly.

Power of a Point Concept

To classify the point, we use its power with respect to the circle. This algebraic quantity can be positive, zero, or negative. An ordinary distance is always zero or positive. For a circle with center $(a,b)$ and radius $r$:

For point $A(x_0, y_0)$ and the general circle $x^2 + y^2 + Dx + Ey + F = 0$, the power of $A$ is:

Substituting the coordinates once gives the value $K_A$. The sign of K_A determines the point's location:

Three Possible Point Positions

Based on the power of a point value, there are three possible positions:

1. The point is inside the circle when $K_A < 0$.

2. The point is on the circle when $K_A = 0$.

3. The point is outside the circle when $K_A > 0$.

The three cases are shown in the graph below:

How to Determine Point Position

Identify the circle equation and the point coordinates, then substitute the coordinates into the power-of-a-point formula to obtain $K_A$.

The sign of $K_A$ determines the point's position. If $K_A<0$, the point is inside the circle. If $K_A=0$, the point is on the circle. If $K_A>0$, the point is outside the circle.

For point $A(1, -2)$ and circle $x^2 + y^2 = 25$, first compute the squared distance to the center. Then subtract $r^2$:

Because $K_A = -20 < 0$, point $A(1, -2)$ lies inside the circle.

For $x^2 + y^2 = r^2$, comparing $d_A^2$ with $r^2$ is equivalent to checking the sign of $K_A=d_A^2-r^2$.

Application for General Form

For a circle in general form $x^2 + y^2 + Dx + Ey + F = 0$, calculate $K_A=x_0^2+y_0^2+Dx_0+Ey_0+F$. The sign of $K_A$ determines whether the point is inside, on, or outside the circle.

For example, for point $A(1, -2)$ and circle $x^2 + y^2 - 8x - 2y - 8 = 0$:

Since $K_A = -7 < 0$, then point $A(1, -2)$ is located inside the circle.

Practice Problems

Each problem gives a point and a circle, and the answer is whether the point lies inside, on, or outside the circle. Compare the distance from the centre with the radius.

1. Determine the position of point $A(3, 4)$ relative to circle $x^2 + y^2 = 16$.

2. Investigate whether point $B(6, -8)$ lies on circle $x^2 + y^2 = 100$.

3. Determine the position of point $C(2, 1)$ relative to circle $x^2 + y^2 - 4x + 6y - 12 = 0$.

4. A circle has equation $(x-3)^2 + (y+2)^2 = 25$. Determine the position of point $D(7, 2)$ relative to that circle.

Worked Solutions

1. Solution:

   Substitute coordinates of point $A(3, 4)$ into circle equation $x^2 + y^2 = 16$:

   $$d_A^2 = 3^2 + 4^2 = 9 + 16 = 25$$

   $$K_A = d_A^2-r^2 = 25-16 = 9$$

   Since $K_A = 9 > 0$, point $A(3, 4)$ lies outside the circle.

2. Solution:

   Substitute coordinates of point $B(6, -8)$ into circle equation $x^2 + y^2 = 100$:

   $$d_B^2 = 6^2 + (-8)^2 = 36 + 64 = 100$$

   $$K_B = d_B^2-r^2 = 100-100 = 0$$

   Since $K_B = 0$, point $B(6, -8)$ lies exactly on the circle.

3. Solution:

   Substitute coordinates of point $C(2, 1)$ into circle equation $x^2 + y^2 - 4x + 6y - 12 = 0$:

   $$K_C = 2^2 + 1^2 - 4(2) + 6(1) - 12$$

   $$K_C = 4 + 1 - 8 + 6 - 12 = -9$$

   Since $K_C = -9 < 0$, then point $C(2, 1)$ is located inside the circle.

4. Solution:

   For circle $(x-3)^2 + (y+2)^2 = 25$ with center $(3, -2)$ and radius $r = 5$.

   Substitute point $D(7, 2)$:

   $$\begin{aligned} d_D^2 &= (7-3)^2 + (2-(-2))^2 \\ &= 4^2 + 4^2 \end{aligned}$$

   $$d_D^2 = 16 + 16 = 32$$

   $$K_D = d_D^2-r^2 = 32-25 = 7$$

   Since $K_D = 7 > 0$, point $D(7, 2)$ lies outside the circle.