CIRCLES

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

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

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)

(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.

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.

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)

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

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).

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

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

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.

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

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

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.

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

SPHERE

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

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

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.

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.

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

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

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