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## CIRCLES

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**Circle**radius Center GEOMETRICAL DEFINITION OF A CIRCLE A circle is the set of all points in a plane equidistant from a fixed point called the center.**-7**-2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 The standard form of the equation of a circle with its center at the origin and radius r is Center at (0, 0)**1**3 5 7 -7 -2 -1 -6 -5 -4 -3 0 4 6 8 2 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). (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.**In other words, a line that intersects the circle in**exactlyone 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 P(x1,y1)**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**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