**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)

(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

**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**

**8. Condition of tangency**

**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

**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

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

**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

**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

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 =

**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**

**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

✅ 12th grade math formulas ⭐️⭐️⭐️⭐️⭐

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