Circle Formula Sheet

A circle is a geometric shape consisting of all points in a plane that are at a constant distance, called the radius, from a fixed point, known as the center. The circumference is the boundary of the circle, and the diameter is twice the radius, which is the longest distance across the circle through its center. Circles are characterized by their perfect symmetry and are a fundamental shape in geometry.

Neetesh Kumar | May 06, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition
• 2. Standard Equations of the Circle
• 3. Position of a Point W.R.T Circle
• 4. Tangent Line of Circle
• 5. Normal of Circle
• 6. Chord of Contact
• 7. Equation of the Chord with a Given Middle Point
• 8. Director Circle
• 9. Pole and Polar
• 10. Family of Circles
• 11. Direct and Transverse Common Tangents
• 12. The Angle of Intersection of Two Circles
• 13. Radical Axis of the Two Circles
• 14. Radical Centre

1. Definition:

A circle is the locus of a point that moves in a plane in such a way that its distance from a fixed point remains constant. The fixed point is called the center of the circle, and the constant distance is called the radius of the circle.

2. Standard Equations of the Circle:

(a) Central Form:
If $(h, k)$ is the center and $r$ is the radius of the circle, then its equation is: $(x – h)^2 + (y – k)^2 = r^2$

(b) General Equation of Circle:
$x^2 + y^2 + 2gx + 2fy + c = 0$ where $g, f, c$ are constants and the centre is $(-g, -f)$ = $(\frac{- Coeff. \space of \space X}{2}, \frac{- Coeff. \space of \space Y}{2})$
Note: The general quadratic equation in x and y, $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ represents a circle if :

• coefficient of $x^2$ = coefficient of $y^2$ or a = b $\ne$ 0
• coefficient of xy = 0 or h =0
• $(g^2 + f^2 - c) \ge 0$ (for a real circle)

(c) Intercepts cut by the circle on axes:
The intercepts cut by the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ on:

• x-axis: $\sqrt{2g - c}$
• y-axis: $\sqrt{2f - c}$

(d) Diameter Form of Circle: If $A(x_1, y_1)$ and $B(x_2, y_2)$ are the endpoints of the diameter of the circle, then the equation of the circle is: $(x – x_1)(x – x_2) + (y – y_1)(y – y_2) = 0$

(e) The Parametric Forms of the Circle:

• The parametric equation of the circle $x^2 + y^2 = r^2$ is: $x = r \cos \theta, \ y = r \sin \theta$

• The parametric equation of the circle $(x – h)^2 + (y – k)^2 = r^2$ is: $x = h + r \cos \theta, \ y = k + r \sin \theta$

3. Position of a Point W.R.T Circle:

(a) Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$ and the point be $(x_1, y_1)$. The point $(x_1, y_1)$ lies outside, on, or inside the circle according as: $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c >, =, < 0$

(b) The greatest and least distance of point A from a circle with center C and radius r is $AC + r$ and $|AC - r|$, respectively.

(c) The power of point is given by $S_1$.

4. Tangent Line of Circle:

When a straight line meets a circle at two coincident points, it is called the tangent of the circle.

(a) Condition of Tangency: The line $L = 0$ touches the circle $S = 0$ if the length of the perpendicular from the centre to that line equals the radius of the circle.

(b) Equation of the Tangent:

• Tangent at the point $(x_1, y_1)$ on the circle $x^2 + y^2 = a^2$ is: $xx_1 + yy_1 = a^2$

• Tangent at the point $(x_1, y_1)$ on the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is: $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$

5. Normal of Circle:

Normal at a point of the circle is the straight line perpendicular to the tangent at the point of contact and passes through the circle's center.

(a) Equation of Normal:

• At the point $(x_1, y_1)$ of the circle $x^2 + y^2 + 2gx + 2fy + c = 0$: $y - y_1 = \frac{y_1 + f}{x_1 + g}(x - x_1)$

• On the circle $x^2 + y^2 = a^2$ at point $(x_1, y_1)$: $y - y_1 = -\frac{x_1}{y_1}(x - x_1)$

6. Chord of Contact:

If two tangents $PT_1$ and $PT_2$ are drawn from point $P(x_1, y_1)$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$, then the equation of the chord of contact $T_1T_2$ is: $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$

7. Equation of the Chord with a Given Middle Point:

The equation of the chord of the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ with mid-point $M(x_1, y_1)$ is: $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$

8. Director Circle:

The locus of the point of intersection of two perpendicular tangents to a circle is called the director circle.

For the circle $x^2 + y^2 = a^2$, the equation of the director circle is: $x^2 + y^2 = 2a^2$

9. Pole and Polar:

Let any straight line through the given point $A(x_1, y_1)$ intersect the given circle $S = 0$ in two points P and Q. If the tangents to the circle at P and Q meet at point R, then the locus of point R is called the polar of the point A, and point A is called the pole.

The equation of the polar of point $(x_1, y_1)$ with respect to circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is: $xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$

10. Family of Circles:

(a) The equation of the family of circles passing through the points of intersection of two circles $S_1 = 0$ and $S_2 = 0$ is: $S_1 + KS_2 = 0$

(b) The equation of the family of circles passing through the point of intersection of a circle $S = 0$ and a line $L = 0$ is given by: $S + KL = 0$

(c) The equation of a family of circles passing through two given points $(x_1, y_1)$ and $(x_2, y_2)$ can be written as: $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) + K = 0$

11. Direct and Transverse Common Tangents:

Let two circles have centers $C_1$ and $C_2$ and radii $r_1$ and $r_2$. The distance between their centers is $C_1C_2$.

(a) Both circles will touch externally if $C_1C_2 = r_1 + r_2$. There are three common tangents.

(b) Both circles will touch internally if $C_1C_2 = |r_1 - r_2|$. There is one common tangent.

(c) The circles will intersect if $|r_1 - r_2| < C_1C_2 < r_1 + r_2$. There are two common tangents.

(d) The circles will not intersect if $C_1C_2 < |r_1 - r_2|$ or $C_1C_2 > r_1 + r_2$. There are no common tangents.

12. The Angle of Intersection of Two Circles:

The angle between the tangents of two circles at the point of intersection of the circles is called the angle of intersection of two circles.

The formula for the cosine of the angle of intersection is: $\cos \theta = \frac{2g_1g_2 + 2f_1f_2 - c_1 - c_2}{\sqrt{(g_1^2 + f_1^2 - c_1)(g_2^2 + f_2^2 - c_2)}}$

13. Radical Axis of the Two Circles:

The locus of a point, which moves in such a way that the length of tangents drawn from it to the circles are equal, is called the radical axis.

For two circles: $S_1: x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ $S_2: x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ The equation of the radical axis is: $S_1 - S_2 = 0$

14. Radical Centre:

The radical center of the three circles is the point from which the length of tangents to the three circles are equal, i.e., the point of intersection of the radical axes of the circles.

Note: If three circles are drawn on the sides of a triangle as diameters, their orthocenter will be the radical center.