# Circle Formulas | Properties of Circle Formulae

Are you fed up with the traditional process of solving Circle problems? then, here is the best way to solve the lengthy circle problems in an easy way. Use direct circle formulas and end up your complex calculations effortlessly at a faster pace. Make use of the required circle formula from the list provided here & understand the concept thoroughly.

## List of Circle Formulae

Memorize the formulas of all circle concepts by using the list of various properties of circle formulae provided over here. With this list of circle formulas, you can easily learn and solve Director Circle, Diameter of a Circle, and many other lengthy circle concepts problems with ease.

**1. General equation of a Circle**

x^{2} + y^{2} + 2gx + 2fy + c = 0

(i) Centre of a general equation of a circle is (-g, -f)

i.e. (-\(\frac{1}{2}\) coefficient of x, –\(\frac{1}{2}\) coefficient of y)

(ii) Radius of a general equation of a circle is \(\sqrt{g^{2}+f^{2}-c}\)

**2. Central form of equation of a Circle**

(i) The equation of a circle having centre (h, k) and radius r, is (x – h)^{2} + (y – k)^{2} = r^{2}

(ii) If the centre is origin then the equation of a circle is x^{2} + y^{2} = r^{2}

**3. Diametral Form**

If (x_{1}, y_{1}) and (x_{2}, y_{2}) be the extremities of a diameter, then the equation of circle is (x – x_{1}) (x – x_{2}) + (y – y_{1}) (y – y_{2}) = 0

**4. The parametric equations of a Circle**

- The parametric equations of a circle x
^{2}+ y^{2}= r^{2}are x = r cos θ, y = r sin θ. - The parametric equations of the circle (x – h)
^{2}+ (y – k)^{2}= r^{2}are x = h + r cos θ, y = k + r sin θ. - Parametric equations of the circle x
^{2}+ y^{2}+ 2gx + 2fy + c = 0 are x = -g + \(\sqrt{g^{2}+f^{2}-c}\) cos θ, y = – f + \(\sqrt{g^{2}+f^{2}-c}\) sin θ

**5. Position of a Point with respect to a Circle**

The following formulae are also true for Parabola and Ellipse.

S_{1} > 0 ⇒ Point is outside the circle.

S_{1} = 0 ⇒ Point is on the circle.

S_{1} < 0 ⇒ Point is inside the circle.

**6. Length of the intercept made by the circle on the line**

p = length of ⊥ from centre to interscenting lines \(=2 \sqrt{r^{2}-p^{2}}\)

**7. The length of the intercept made by line y = mx + c with the circle**

x^{2} + y^{2} = a^{2} is \(2 \sqrt{\frac{\mathrm{a}^{2}\left(1+\mathrm{m}^{2}\right)-\mathrm{c}^{2}}{1+\mathrm{m}^{2}}}\)

**8. Condition of tangency**

(a) Standard Case: Circle x^{2} + y^{2} = a^{2} will touch the line y = mx + c if c = ± \(a \sqrt{1+m^{2}}\)

(b) General Case: If m is slope of line, circle is

x^{2} + y^{2} + 2gx + 2fy + c = 0 then condition is

y + f = m(n + g) ±\(\sqrt{g^{2}+f^{2}-c}\)

**9. Intercepts made on coordinate axes by the circle**

- x-axis = 2\(\sqrt{g^{2}-c}\)
- y-axis = 2\(\sqrt{f^{2}-c}\)

**10. Equation of Tangent T = 0**

(i) The equation of tangent to the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 at a point (x_{1} y_{1}) is xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0

(ii) The equation of tangent to circle x^{2} + y^{2} = a^{2} at point (x_{1}, y_{1}) is xx_{1} + yy_{1} = a^{2}

(iii) Slope Form:

From condition of tangency for every value of m, the line y = mx ± a\(\sqrt{1+m^{2}}\) is a tangent of the circle x^{2} + y^{2} = a^{2} and its point of contact is \(\left(\frac{\mp \mathrm{am}}{\sqrt{1+\mathrm{m}^{2}}}, \frac{\pm \mathrm{a}}{\sqrt{1+\mathrm{m}^{2}}}\right)\)

**11. Equation of Normal**

- The equation of normal to the circle x
^{2}+ y^{2}+ 2gx + 2fy + c = 0 at any point (x_{1}, y_{1}) is y – y_{1}= \(\frac{y_{1}+f}{x_{1}+g}\)(x – x_{1}) - The equation of normal to the circle x
^{2}+ y^{2}= a^{2}at any point (x_{1}, y_{1}) is xy_{1}– x_{1}y = 0

**12. Length of tangent \(\sqrt{\mathrm{S}_{1}}\) = AC = AB**

**13. Pair of tangents SS _{1} = T^{2}**

**14. Chord of contact T = 0**

T = xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0

- The length of chord of contact = 2\(\sqrt{r^{2}-p^{2}}\)
- Area of AABC is given by \(\frac{a\left(x_{1}^{2}+y_{1}^{2}-a^{2}\right)^{3 / 2}}{x_{1}^{2}+y_{1}^{2}}\)

**15. Director Circle**

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

Let the circle be x^{2} + y^{2} = a^{2}, the equation of director circle is x^{2} + y^{2} = 2a^{2}, director circle is a concentric circle whose radius is \(\sqrt{2}\) times the radius of the given circle.

**16. Equation of Polar and coordinates of Pole**

- Equation of polar is T = 0
- Pole of polar Ax + By + C = 0 with respect to circle x
^{2}+ y^{2}= a^{2}is \(\left(-\frac{\mathrm{Aa}^{2}}{\mathrm{C}},-\frac{\mathrm{Ba}^{2}}{\mathrm{C}}\right)\)

17. Equation of a chord whose middle point is given T = S_{1}

18. The equation of the circle passing through the points of intersection of the circle S = 0 and line **L = 0 is S + λL = 0.**

**19. Diameter of a circle**

The diameter of a circle x^{2} + y^{2} = r^{2} corresponding to the system of parallel chords y = mx + c is x + my = 0.

**20. Equation of common chord S _{1} – S_{2} = 0**

**21. Two circles with radii r _{1}, r_{2}** and d be the distance between their centres then the angle of intersection θ between them is given by cos θ = \(\frac{r_{1}^{2}+r_{2}^{2}-d^{2}}{2 r_{1} r_{2}}\)

**22. Condition of Orthogonality**

2g_{1}g_{2} + 2f_{1}f_{2} = c_{1} + c_{2}

**23. Relative position of two circles and No. of common tangents**

Let C_{1} (h_{1}, k_{1}) and C_{2} (h_{2}, k_{2}) be the centre of two circle and r_{1}, r_{2} be their radius then

- C
_{1}C_{2}> r_{1}+ r_{2}⇒ do not intersect or one outside the other ⇒ 4 common tangents - C
_{1}C_{2}< |r_{1}– r_{2}| ⇒ one inside the other => 0 common tangent - C
_{1}C_{2}= r_{1}+ r_{2}⇒ external touch ⇒ 3 common tangents - C
_{1}C_{2}= |r_{1}– r_{2}| ⇒ internal touch ⇒ 1 common tangent - |r
_{1}– r_{2}| < C_{1}C_{2}< r_{1}+ r_{2}⇒ intersection at two real points ⇒ 2 common tangents

**24. Equation of the common tangents at point of contact S _{1} – S_{2} = 0.**

**25. Pair of point of contact**

The point of contact divides C_{1}C_{2} in the ratio r_{1}: r_{2} internally or externally as the case may be.

**26. Radical axis and radical center**

- Definition of Radical axis: Locus of a point from which length of tangents to the circles are equal, is called radical axis.
- radical axis is S – S’ = 0
- If S
_{1}= 0, S_{2}= 0 and S_{3}= 0 be any three given circles then the radical centre can be obtained by solving any two of the following equations

S_{1}– S_{2}= 0, S_{2}– S_{3}= 0, S_{3}– S_{1}= 0.

**27. S _{1} – S_{2} = 0 represent equation of all i.e. Radical axis, common axis, common tangent i.e.**

when circle are not in touch → Radical axis

when circle are in touch → Common tangent

when circle are intersecting → Common chord

**28. Let θ _{1} and θ_{2} are two points lies on circle x^{2} + y^{2} = a^{2}, then equation of line joining these two points is**

\(\mathrm{x} \cos \left(\frac{\theta_{1}+\theta_{2}}{2}\right)+\mathrm{y} \sin \left(\frac{\theta_{1}+\theta_{2}}{2}\right)=\mathrm{a} \cos \left(\frac{\theta_{1}-\theta_{2}}{2}\right)\)

**29. Limiting Point of co-axial system of circles:**

Limiting point of a system of co-axial circles are the centres of the point circles belonging to the family. Two such point of a co-axial are (± \(\sqrt{\mathrm{c}}\), 0).