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CIRCLES

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**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

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**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)

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**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)

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**(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.

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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.

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**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)

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

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**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).

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**Length of the tangent from the point P(x1,y1) to the circle **

x2 + y2 + Dx + Ey + F = 0 is (x1,y1)

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**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

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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.

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

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**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

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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.

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**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

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SPHERE

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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

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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

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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.

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**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.

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

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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

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

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