1. CIRCLES

2. GEOMETRICAL DEFINITION OF A CIRCLE
A circle is the set of all points in a plane equidistant from a fixed point called the center.
Center
Circle
radius

3. The standard form of the equation of a circle with its center at the origin and radius r is2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
Center at (0, 0)

4. The center of the circle is at (h, k).
If the center of the circle is NOT at the origin then the equation for the standard form of a circle looks like:
The center of the circle is at (h, k). 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
(h, k)

5. (2) General equation of a circle is
x2 + y2 + Dx + Ey + F = 0
Note: The given quadratic relation will be a circle if the coefficients of the x2 term and y2 term are equal and the xy term is zero.

6. Tangent to the circle.
Let A be a fixed point on the circumference of circle O and P be another variable point on the circumference. As P approaches A along the circumference, the chord AP will rotate about A. The limiting position AT of the variable chord AP is called the tangent to the circle O at the point A and A is the point of contact.

7. Equation of the tangent to the circle x2 + y2 + Dx + Ey + F = 0
In other words, a line that intersects the circle in exactly one point is said to be tangent to the circle. .
Equation of the tangent to the circle
x2 + y2 + Dx + Ey + F = 0
at the point P(x1,y1)

8. slope = m
The equations of the two tangents with slope m to the circle x2 + y2 = r2 are

9. Condition for tangency
A straight line
y = mx + c
is a tangent to the circle
x2 + y2 = r2
if and only if
c2 = r2( 1 + m2).

10. Length of the tangent from the point P(x1,y1) to the circle
x2 + y2 + Dx + Ey + F = 0 is
(x1,y1)

11. Common chord b\w two circles.
Common chord / tangent of two circles
x2 + y2 + D1x + E1y + F1= 0
and x2 + y2 + D2x + E2y + F2 = 0
is given by:
(D1 - D2)x + (E1 - E2)y + (F1 - F2)=0

12. Normal to the circle
Let P be a point on the circumference if circle O. A straight line PN passing through P and being perpendicular to the tangent PT at P is called the normal to the circle O at P.

13. The equations of the normal to the circle x2 + y2 + Dx + Ey + F = 0 at the point P(x1, y1) is

14. Circles passing through the intersection
of the circle x2 + y2 + Dx + Ey + F = 0
and the straight line Ax + By + C =0
x2 + y2 + Dx + Ey + F +k(Ax + By + C)= 0
Family of circles passing through
the intersection of the two circles
x2 + y2 + D1x + E1y + F1= 0
and x2 + y2 + D2x + E2y + F2 = 0
x2 + y2 + D1x + E1y + F1 + k(x2 + y2 + D2x + E2y + F2)= 0

15. The Chord of Contact
Let P be a point lying outside a circle. PA, PB are two tangents drawn to the circle from P touching the circle at A and B respectively. The chord AB joining the points of contact is called the chord of contact of tangents drawn to the circle from an external point P.

16. Equation of the Chord of Contact
Let P(x1, y1) be a point lying outside the circle
x2 + y2 + Dx + Ey + F = 0. Then the equation

17. SPHERE

18. The sphere appears in nature whenever a surface wants to be as small as possible. Examples include bubbles and water drops. Some special spheres in nature are
BRASS SPHERE
DARK SPHERE
JADE SPHERE
LIGHT SPHERE

19. A sphere is defined as the set of all points in three-dimensional space that are located at a distance r (the "radius") from a given point (the "center"). Equation of the sphere with center at the origin (0,0,0) and radius R is given by
The Cartesian equation of a sphere centered at the point (x0,y0,z0) with radius R is given by

20. GREAT CIRCLE
A great circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as distinct from a small circle. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, and have the same center as the sphere. A great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean space is a great circle of exactly one sphere.

21. Planes through a sphere
A plane can intersect a sphere at one point in which case it is called a tangent plane. Otherwise if a plane intersects a sphere the "cut" is a circle. Lines of latitude are examples of planes that intersect the Earth sphere.

22. The intersection of the spheres is a curve lying in a plane which is a circle with radius r.

23. Sphere Facts
It is perfectly symmetrical
It has no edges or vertices (corners)
It is not a polyhedron
All points on the surface are the same distance from the center

24. THE END BY
KUNJAN GUPTA
MATH DEPTT.
PGGCG-11, CHD