Sunday, January 17, 2016

Circle - Chapter Revision Points

Sections in the chapter

1. Definition
2. Standard equation of a circle
3. Some particular cases of the central form of the equation of a circle
4. General equation of a circle
5. Equation of a circle when the co-ordinates of end points of a diameter are given
6. Intercepts on the axis
7. Position of a point with respect to a circle
8. Equation of a circle in parametric form
9. Intersection of a straight line and a circle
10. The length of the intercept cut off from a line by a circle
11. Tangent to a circle at a given point.
12. Normal to a circle at a given point
13. Length of the tangent from a point to a circle
14. Pair of tangents drawn from a point to a given circle
15. Combined equation of pair of tangents
16. Director circle and its equation
17. Chord of contact of tangents
18. Pole and polar
19. Equation of the chord bisected at a given point
20. Diameter of a circle
21. Common tangents to two circles
22. Common chord of two circles
23. Angle of intersection of two curves and the condition of orthogonality of two circles.
24. Radical axis
25. Equation of a circle through the intersection of a circle and line
26. Circle through the intersection of two circles
27. Coaxial system of circles

Revision Points

Equation of circle in various forms

a. Centre (h,k) and radius a

b. Centre (h,k) and passing through origin

c. Centre (h,k) and circle touches the axis of x

d. Centre (h,k) and circle touches the axis of y

e. Centre (h,k) and and circle touches both axes.

f. general equation

g. circle whose diameter is the line joining two points

h. Circle through three given points

i. Parametric equation of the circle

h. equation of a circle that touches given circle at a given point

j. Equation of a circle passing through the intersection of given circles S1 = 0 and S2 = 0

The standard equation of a circle with center C(h,k) and radius r is as follows:

(x - h)² + (y - k)² = r²

Parametric Equations
The equation of a circle, centred at the origin, is: x2 + y2 = a2, where a is the radius.

Suppose we have a curve which is described by the following two equations:

x = acosθ (1)
y = asinθ (2)

We can eliminate q by squaring and adding the two equations:

x² + y² = a²cos²θ + a²sin²θ = a² .

Hence equations (1) and (2) together also represent a circle centred at the origin with radius a and are known as the parametric equations of the circle. θ is known as the parameter. As θ varies between 0 and 2π, x and y vary.

Updated  17 Jan 2016, 28 April 2008

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