Power of a point wrt Circle: Meaning, Theorem, Formula

Power of a point wrt Circle

Edited By Komal Miglani | Updated on Jul 02, 2025 07:52 PM IST

The concept of the "power of a point" relative to a circle is a fascinating and useful idea in geometry. It provides a relationship between a point and a circle that can be used in various geometric proofs and constructions. This concept extends beyond basic circle properties, offering deeper insights into the interactions between points and circles.

This Story also Contains

Equation of circle
Power of a point wrt Circle
Solved Examples Based on Power of a Point wrt Circle

Power of a point wrt Circle

Equation of circle

A circle is the locus of a moving point such that its distance from a fixed point is constant.

The fixed point is called the centre (O) of the circle and the constant distance is called its radius (r)

The equation of a circle with centre at C (h,k) and radius r is (x - h)²+ (y - k)²= r²

Let P(x, y) be any point on the circle. Then, by definition, $|C P|=r$

Using the distance formula, we have

$\begin{array}{ll} & \sqrt{(x-h)^2+(y-k)^2}=r \text { i.e. } \quad(x-h)^2+(y-k)^2=r^2\end{array}$

If the centre of the circle is the origin or (0,0) then the equation of the circle becomes

$\begin{aligned} & (x-0)^2+(y-0)^2=r^2 \\ & \text { i.e. } x^2+y^2=r^2\end{aligned}$

Power of a point wrt Circle

The power of a point $\mathrm{P}(\mathrm{a}, \mathrm{b})$ with respect to the circle $S: \mathrm{x}^2+\mathrm{y}^2+2 g \mathrm{~g}+2 \mathrm{fy}+\mathrm{c}=0$ is $\mathrm{S}_1$, where $\mathrm{S}_1: \mathrm{a}^2+\mathrm{b}^2+2 \mathrm{ga}+2 \mathrm{fb}+\mathrm{c}=0$

We know $\mathrm{PA} \cdot \mathrm{PB}=(\mathrm{PT})^2$
Also we know that $\mathrm{PT}=\sqrt{S_1}$
So, $\mathrm{PA} \cdot \mathrm{PB}=\mathrm{PT}^2=\mathrm{S}_1$

Chord of Contact

S is a circle and P(x₁,y₁) be an external point to a circle S. A and B are the points of contact of the tangents drawn from P to circle S. Then the chord AB is called the chord of contact of the circle S drawn from an external point P.

To get the equation of the chord of contact of external point $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ with respect to the circle $x^2+y^2+2 g x+2 f y+c=0$, we use the formula $\mathrm{T}=0$
So the equation of chord of contact is $x x_1+y y_1+g\left(x+x_1\right)+f\left(y+y_1\right)+c=0$ mathematician's toolkit.

Solved Examples Based on Power of a Point wrt Circle

Example 1: Find the length of the tangent from Point $\mathrm{P}(0,0)$ on the circle $2 x^2+2 y^2+8 x-8 y+8=0$

1) 2

2) $2 \sqrt{2}$

3) 4

4) 8

Solution

The power of a point $\mathrm{P}(\mathrm{a}, \mathrm{b})$ with respect to the circle $S: \mathrm{x}^2+\mathrm{y}^2+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c}=0$ is $\mathrm{S}_1$, where $\mathrm{S}_1: \mathrm{a}^2+\mathrm{b}^2+2 \mathrm{ga}+2 \mathrm{fb}+\mathrm{c}=0$.
$\mathrm{PA} \cdot \mathrm{PB}=(\mathrm{PT})^2=\mathrm{S}_1$
From above concept
length of tangent $=\sqrt{\left(S_1\right)}$
Remember: Factor of $x^2$ is 1
Given $2 x^2+2 y^2+8 x-8 y+8=0$
$\Rightarrow x^2+y^2+4 x-4 y+4=0$
length of tangent $=\sqrt{\left(S_1\right)}=\sqrt{4}=2$

Example 2: A line from point $\mathrm{P}(2,-1)$ to the circle $x^2+y^2+4 x+4 y+4=0$ intersects it at A and B, then the value of $P A \cdot P B$ is

1) 12

2) 3

3) 16

4) None of these

Solution

We know

$\begin{aligned}
& P A \cdot P B=S_1 \\
& S_1=(2)^2+(-1)^2+4 \times 2+4 \times-1+4 \\
& P A \cdot P B=13
\end{aligned}$
Hence, the answer is the option 2.

Example 3: Length of a tangent from a point $(-5,-4)$ to the circle $x^2+y^2-4 x+2 x-10=0$ is

1) 7
2) $\sqrt{48}$
3) 9
14) $\sqrt{43}$

Solution
Length of tangent $=\sqrt{(-5)^2+(-4)^2-4(-5)+2(-4)-10}=\sqrt{43}$
Hence, the answer is the option 4.

Example 4: A variable circle C has the equation
$x^2+y^2-2\left(t^2-3 t+1\right) x-2\left(t^2+2 t\right) y+t=0$, where $t$ is a parameter.
If the power of point $P(a, b)$ w.r.t. the circle $C$ is constant then the ordered pair $(a, b)$ is:

1) $\left(\frac{1}{10},-\frac{1}{10}\right)$
2) $\left(-\frac{1}{10}, \frac{1}{10}\right)$
3) $\left(\frac{1}{10}, \frac{1}{10}\right)$
4) $\left(-\frac{1}{10},-\frac{1}{10}\right)$

Solution
Power:

$\begin{aligned}
P & =a^2+b^2-2\left(t^2-3 t+1\right) a-2\left(t^2+2 t\right) b+t \\
& =-(2 a+2 b) t^2+(6 a-4 b+1) t+a^2+b^2-2 a
\end{aligned}$

This power is independent of the parameter $t$ if and only if: $2 a+2 b=0 \quad \Rightarrow \quad a=-b$
and $6 \mathrm{a}-4 \mathrm{~b}+1=0$
$\Rightarrow \quad \mathrm{a}=-\frac{1}{10}$ and $\mathrm{b}=\frac{1}{10}$
Hence, the answer is the option (2).

Example 5: Polar of origin $(0,0)$ w.r.t. the circle $\mathrm{x}^2+\mathrm{y}^2+2 \lambda \mathrm{x}+2 \mu \mathrm{y}+\mathrm{c}=0$ touches the circle $\mathrm{x}^2+\mathrm{y}^2=\mathrm{r}^2$, if

1) $\mathrm{c}=r\left(\lambda^2+\mu^2\right)$
2) $\mathrm{r}=\mathrm{c}\left(\lambda^2+\mu^2\right)$
3) $\mathrm{c}^2=\mathrm{r}^2\left(\lambda^2+\mu^2\right)$
4) $\mathrm{r}^2=\mathrm{c}^2\left(\lambda^2+\mu^2\right)$

Solution

$x^2+y^2+21 x+2 m y+c=0$

Polar of $(0,0)$ is $x .0+0 . y+\lambda(x+0)+\mu(y+0)+c=0$

$\lambda x+\mu y+c=0 \quad \ldots(1)$

(1) will touch the circle $x^2+y^2=r^2$ if the distance of origin from (1) $=r$

$\frac{|0+0 \times \mathrm{c}|}{\sqrt{\mathrm{I}^2+\mathrm{m}^2}}=\mathrm{r} \quad \mathrm{p} \quad \mathrm{c}^2=\mathrm{r}^2\left(\mathrm{I}^2+\mathrm{m}^2\right)$

Hence, the answer is the option (2).

Frequently Asked Questions (FAQs)

1. What is the power of a point with respect to a circle?

The power of a point with respect to a circle is a measure of how far the point is from the circle. It's defined as the product of the distances from the point to the two points where any line through the point intersects the circle. This value remains constant for all such lines through the point.

2. How is the power of a point calculated when the point is outside the circle?

For a point outside the circle, the power is calculated as the square of the length of the tangent from that point to the circle. It's also equal to d^2 - r^2, where d is the distance from the point to the circle's center, and r is the radius of the circle.

3. What does a negative power of a point indicate?

A negative power indicates that the point is inside the circle. The magnitude of the negative power represents how far inside the circle the point is located.

4. How does the power of a point change as the point moves closer to or farther from the circle?

As a point moves closer to the circle from the outside, its power decreases. When the point is on the circle, its power is zero. As the point moves inside the circle, its power becomes negative and increases in magnitude as it approaches the center.

5. What is the relationship between the power of a point and the radical axis of two circles?

The radical axis of two circles is the locus of points that have equal power with respect to both circles. It's a straight line perpendicular to the line joining the centers of the two circles.

6. How can the power of a point be used to determine if a line is tangent to a circle?

If a line passes through a point P and intersects a circle at points A and B, the line is tangent to the circle if and only if PA * PB = 0. This is because the power of a point on the circle with respect to that circle is zero.

7. What is the significance of the power of a point being positive, negative, or zero?

A positive power indicates the point is outside the circle, a negative power means the point is inside the circle, and a power of zero means the point lies exactly on the circle's circumference.

8. How is the power of a point related to the concept of inversion in a circle?

In circle inversion, the power of a point is used to determine the location of its inverse point. If P is a point and O is the center of inversion, then OP * OP' = k^2, where P' is the inverse of P and k is the radius of the circle of inversion.

9. Can two different points have the same power with respect to a circle?

Yes, two different points can have the same power with respect to a circle. These points would lie on a circle concentric with the original circle.

10. How does the power of a point relate to the radical center of three circles?

The radical center is the point where the radical axes of all pairs of three circles intersect. This point has equal power with respect to all three circles.

11. What is the power of the center of a circle with respect to that circle?

The power of the center of a circle with respect to that circle is equal to the negative of the square of the radius (-r^2). This is because the center is inside the circle at a distance equal to the radius from the circumference.

12. How can the power of a point be used to construct tangent lines to a circle?

To construct tangent lines from an external point P to a circle, we can use the fact that the power of P equals the square of the length of the tangent. This allows us to determine the tangent points on the circle.

13. What is the relationship between the power of a point and the length of a chord?

If a line through a point P intersects a circle at points A and B, forming a chord AB, then the power of P with respect to the circle is equal to PA * PB, which is also equal to the square of half the length of the chord if P is the midpoint of the chord.

14. How does the concept of power of a point extend to spheres in three-dimensional space?

The power of a point with respect to a sphere is defined similarly to that of a circle. It's the product of the distances from the point to any two points where a line through the point intersects the sphere. The concept of power planes (analogous to radical axes) also exists for spheres.

15. Can the power of a point be used to solve problems involving intersecting or tangent circles?

Yes, the power of a point is a powerful tool for solving problems involving intersecting or tangent circles. It can be used to find points of intersection, determine tangency conditions, and calculate distances between circles.

16. What is the geometric interpretation of the power of a point?

Geometrically, the power of a point P with respect to a circle is the square of the length of the tangent from P to the circle if P is outside, zero if P is on the circle, and the negative of the square of half the length of the shortest chord through P if P is inside the circle.

17. How is the power of a point used in the construction of the radical axis of two circles?

The radical axis of two circles is constructed by finding two points that have equal power with respect to both circles. These points can be found by intersecting the circles if they intersect, or by using external or internal tangents if they don't.

18. What is the relationship between the power of a point and the orthogonal circles?

Two circles are orthogonal if they intersect at right angles. The center of each circle has a power equal to the square of its radius with respect to the other circle. This property is used in the construction of orthogonal circles.

19. How does the concept of power of a point relate to the properties of cyclic quadrilaterals?

In a cyclic quadrilateral ABCD, if P is any point on the circumcircle, then PA * PC = PB * PD. This is a direct application of the constant power of a point (in this case, zero) with respect to the circle.

20. Can the power of a point be negative, and if so, what does this mean?

Yes, the power of a point can be negative. A negative power indicates that the point is inside the circle. The magnitude of the negative power represents how far inside the circle the point is located.

21. How is the power of a point used in the theory of coaxal circles?

Coaxal circles are circles whose radical axis is the same line. The power of a point on this radical axis is the same with respect to all circles in the coaxal system, which is a key property in understanding and working with coaxal circles.

22. What is the relationship between the power of a point and the radical center of three circles?

The radical center of three circles is the point where the radical axes of all pairs of the three circles intersect. This point has equal power with respect to all three circles, which is why it lies on all three radical axes.

23. How can the power of a point be used to determine if four points are concyclic?

Four points A, B, C, and D are concyclic (lie on the same circle) if and only if the power of any point P with respect to circle ABC is equal to its power with respect to circle ABD. This is because if the points are concyclic, both these powers would be equal to the power of P with respect to the circle ABCD.

24. What is the significance of the power of a point in the study of pole and polar relationships?

The power of a point is fundamental in defining the polar of a point with respect to a circle. If P is a point and its polar intersects a line through P at Q, then the power of P with respect to the circle equals OP * OQ, where O is the center of the circle.

25. How does the power of a point relate to the concept of inversion in a circle?

26. Can the power of a point be used to solve problems involving tangent circles?

Yes, the power of a point is particularly useful in problems involving tangent circles. For example, if two circles are tangent at a point T, any point P on their radical axis will have PT^2 equal to its power with respect to either circle.

27. How is the power of a point related to the radical axis of two non-intersecting circles?

The radical axis of two non-intersecting circles is the locus of points that have equal power with respect to both circles. It's a straight line perpendicular to the line joining the centers of the two circles, even when the circles don't intersect.

28. What is the relationship between the power of a point and the lengths of tangents drawn from that point to different circles?

If tangents are drawn from a point P to different circles, the difference of the squares of the lengths of these tangents is constant for all circles having the same radical axis. This constant is equal to the difference in the powers of P with respect to these circles.

29. How can the power of a point be used to solve problems involving circles and straight lines?

The power of a point can be used to determine the intersection points of a line and a circle, to find the equation of tangent lines to a circle from an external point, and to solve various other geometric problems involving circles and lines.

30. What is the connection between the power of a point and the concept of radical circles?

Radical circles are circles that have the same radical axis. For any point on this radical axis, its power with respect to all circles in the radical system is the same. This property is fundamental in understanding and working with systems of circles.

31. How does the power of a point change as you move along a line passing through the center of a circle?

As you move along a line passing through the center of a circle, the power of the point increases quadratically as you move away from the circle in either direction. It's zero at the points where the line intersects the circle, and reaches its minimum (negative) value at the center of the circle.

32. Can the power of a point be used to determine the nature of intersection of two circles?

Yes, the power of a point can be used to determine whether two circles intersect, are tangent, or do not intersect. If the power of the center of one circle with respect to the other is less than, equal to, or greater than the square of the sum of their radii, it determines the nature of their intersection.

33. How is the power of a point used in the construction of a tangent to a circle from an external point?

To construct a tangent from an external point P to a circle with center O, we can use the fact that the power of P equals the square of the length of the tangent. This allows us to construct a right triangle where one leg is the radius and the hypotenuse is the line from O to P.

34. What is the relationship between the power of a point and the radical axis of two intersecting circles?

For two intersecting circles, their radical axis passes through their points of intersection. Any point on this radical axis has equal power with respect to both circles, which is why the radical axis is also called the "line of equal power."

35. How can the power of a point be used to solve problems involving circles inscribed in or circumscribed about triangles?

The power of a point can be used to find relationships between the radii of inscribed and circumscribed circles of a triangle, to locate centers of these circles, and to solve various problems related to the geometry of triangles and associated circles.

36. What is the significance of the power of a point in the study of apollonian circles?

Apollonian circles are sets of circles that are tangent to three given circles. The concept of power of a point is crucial in constructing and understanding the properties of Apollonian circles, particularly in determining points of tangency and relationships between radii.

37. How does the power of a point relate to the concept of radical axis in three dimensions?

In three dimensions, the concept of radical axis extends to a radical plane for two spheres. Points on this plane have equal power with respect to both spheres. For three spheres, their radical planes intersect in a line called the radical line.

38. Can the power of a point be used to determine the center and radius of a circle given three points on its circumference?

Yes, the power of a point can be used in this construction. By considering the perpendicular bisectors of two chords (each defined by two of the three given points), we can find the center of the circle. The radius can then be determined using the power of the center point.

39. How is the power of a point related to the concept of inversion circles?

In circle inversion, the power of inversion (k^2) determines the relationship between a point and its inverse. If P is inverted to P' with respect to a circle with center O, then OP * OP' = k^2. This constant k^2 is the power of inversion.

40. What is the relationship between the power of a point and the radical center of three non-coaxal circles?

The radical center of three non-coaxal circles is the point where the radical axes of all pairs of the three circles intersect. This point has equal power with respect to all three circles, which is why it's called the radical center.

41. How can the power of a point be used to solve problems involving tangent circles to two given circles?

The power of a point can be used to determine the centers and radii of circles tangent to two given circles. By considering points with equal power to both given circles (i.e., points on their radical axis), we can locate potential centers for the tangent circles.

42. What is the significance of the power of a point in the study of circle bundles?

In the study of circle bundles (families of circles with a common property), the power of a point is often used to define and analyze relationships between circles in the bundle. It's particularly useful in understanding coaxal systems and other special configurations of circles.

43. How does the concept of power of a point extend to other conic sections?

While the power of a point is primarily defined for circles, similar concepts can be extended to other conic sections. For example, in the case of ellipses and hyperbolas, analogous properties involving products of distances can be defined and studied.

44. Can the power of a point be used to determine if a circle is orthogonal to another circle?

Yes, two circles are orthogonal if and only if the power of the center of each circle with respect to the other is equal to the square of its own radius. This property is fundamental in the study and construction of orthogonal circles.

45. How is the power of a point used in the theory of circle packing?

In circle packing problems, the power of a point can be used to determine relationships between radii of packed circles, to find points of tangency, and to analyze the overall structure of the packing. It's particularly useful in problems involving Apollonian gaskets and related configurations.

46. What is the relationship between the power of a point and the concept of power circles?

Power circles are circles on which every point has the same power with respect to a given circle. They are concentric with the given circle. The concept of power circles is useful in understanding how the power of a point varies in the plane with respect to a fixed circle.

47. How can the power of a point be used to solve problems involving circles and spheres in higher dimensions?

The concept of power of a point extends naturally to higher dimensions. In n-dimensional space, we can define the power of a point with respect to a hypersphere. This generalization is useful in solving geometric problems in higher-dimensional spaces.

48. What is the significance of the power of a point in the study of Steiner chains?

Steiner chains are configurations of circles where each circle in the chain is tangent to two fixed circles and its two neighboring circles in the chain. The power of a point is crucial in analyzing the properties of Steiner chains, particularly in determining the number of circles in the chain and their radii.

49. How does the power of a point relate to the concept of radical circles in three dimensions?

In three dimensions, radical circles become radical spheres. Points with equal power to two given spheres form a plane (the radical plane). The intersection of three such planes (for three spheres) gives a line of points with equal power to all three spheres, analogous to the radical center in two dimensions.

50. Can the power of a point be used to solve problems involving circles and complex numbers?

Yes, the power of a point can be elegantly expressed and manipulated using complex numbers. In the complex plane, circles and lines can be represented by equations involving complex variables, and the power of a point can be calculated and analyzed using complex arithmetic.