2GEOMETRICAL DEFINITION OF A CIRCLE A circle is the set of all points in a plane equidistant from a fixed point called the center.CenterCircleradius
3The standard form of the equation of a circle with its center at the origin and radius r is 2-7-6-5-4-3-2-11573468Center at (0, 0)
4The 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).2-7-6-5-4-3-2-11573468(h, k)
5(2) General equation of a circle is x2 + y2 + Dx + Ey + F = 0Note: 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.
6Tangent 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.
7Equation 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 circlex2 + y2 + Dx + Ey + F = 0at the point P(x1,y1)
8slope = mThe equations of the two tangents with slope m to the circle x2 + y2 = r2 are
9Condition for tangency A straight liney = mx + cis a tangent to the circlex2 + y2 = r2if and only ifc2 = r2( 1 + m2).
10Length of the tangent from the point P(x1,y1) to the circle x2 + y2 + Dx + Ey + F = 0 is(x1,y1)
11Common chord b\w two circles. Common chord / tangent of two circlesx2 + y2 + D1x + E1y + F1= 0and x2 + y2 + D2x + E2y + F2 = 0is given by:(D1 - D2)x + (E1 - E2)y + (F1 - F2)=0
12Normal to the circleLet 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.
13The equations of the normal to the circle x2 + y2 + Dx + Ey + F = 0 at the point P(x1, y1) is
14Circles passing through the intersection of the circle x2 + y2 + Dx + Ey + F = 0and the straight line Ax + By + C =0x2 + y2 + Dx + Ey + F +k(Ax + By + C)= 0Family of circles passing throughthe intersection of the two circlesx2 + y2 + D1x + E1y + F1= 0and x2 + y2 + D2x + E2y + F2 = 0x2 + y2 + D1x + E1y + F1 + k(x2 + y2 + D2x + E2y + F2)= 0
15The Chord of ContactLet 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.
16Equation of the Chord of Contact Let P(x1, y1) be a point lying outside the circlex2 + y2 + Dx + Ey + F = 0. Then the equation
18The 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 areBRASS SPHEREDARK SPHEREJADE SPHERELIGHT SPHERE
19A 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 byThe Cartesian equation of a sphere centered at the point (x0,y0,z0) with radius R is given by
20GREAT CIRCLEA 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.
21Planes 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.
22The intersection of the spheres is a curve lying in a plane which is a circle with radius r.
23Sphere FactsIt is perfectly symmetricalIt has no edges or vertices (corners)It is not a polyhedronAll points on the surface are the same distance from the center