Circle Formulas | Properties 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

x² + y² + 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 = r²

(ii) If the centre is origin then the equation of a circle is x² + y² = r²

3. Diametral Form

If (x₁, y₁) and (x₂, y₂) be the extremities of a diameter, then the equation of circle is (x – x₁) (x – x₂) + (y – y₁) (y – y₂) = 0

4. The parametric equations of a Circle

• The parametric equations of a circle x² + y² = r² are x = r cos θ, y = r sin θ.
• The parametric equations of the circle (x – h)² + (y – k)² = r² are x = h + r cos θ, y = k + r sin θ.
• Parametric equations of the circle x² + y² + 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.
S₁ > 0 ⇒ Point is outside the circle.
S₁ = 0 ⇒ Point is on the circle.
S₁ < 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

x² + y² = a² 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 x² + y² = a² 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
x² + y² + 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 x² + y² + 2gx + 2fy + c = 0 at a point (x₁ y₁) is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

(ii) The equation of tangent to circle x² + y² = a² at point (x₁, y₁) is xx₁ + yy₁ = a²

(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 x² + y² = a² 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 x² + y² + 2gx + 2fy + c = 0 at any point (x₁, y₁) is y – y₁ = \(\frac{y_{1}+f}{x_{1}+g}\)(x – x₁)
• The equation of normal to the circle x² + y² = a² at any point (x₁, y₁) is xy₁ – x₁y = 0

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

13. Pair of tangents SS₁ = T²

14. Chord of contact T = 0

T = xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + 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 x² + y² = a², the equation of director circle is x² + y² = 2a², 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 x² + y² = a² 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 = S₁

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 x² + y² = r² corresponding to the system of parallel chords y = mx + c is x + my = 0.

20. Equation of common chord S₁ – S₂ = 0

21. Two circles with radii r₁, r₂ 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

2g₁g₂ + 2f₁f₂ = c₁ + c₂

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

Let C₁ (h₁, k₁) and C₂ (h₂, k₂) be the centre of two circle and r₁, r₂ be their radius then

• C₁C₂ > r₁ + r₂ ⇒ do not intersect or one outside the other ⇒ 4 common tangents
• C₁C₂ < |r₁ – r₂| ⇒ one inside the other => 0 common tangent
• C₁C₂ = r₁ + r₂ ⇒ external touch ⇒ 3 common tangents
• C₁C₂ = |r₁ – r₂| ⇒ internal touch ⇒ 1 common tangent
• |r₁ – r₂| < C₁C₂ < r₁ + r₂ ⇒ intersection at two real points ⇒ 2 common tangents

24. Equation of the common tangents at point of contact S₁ – S₂ = 0.

25. Pair of point of contact

The point of contact divides C₁C₂ in the ratio r₁: r₂ 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 S₁ = 0, S₂ = 0 and S₃ = 0 be any three given circles then the radical centre can be obtained by solving any two of the following equations
  S₁ – S₂ = 0, S₂ – S₃ = 0, S₃ – S₁ = 0.

27. S₁ – S₂ = 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 θ₁ and θ₂ are two points lies on circle x² + y² = a², 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).

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