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Published byAiden Oxborrow Modified about 1 year ago

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

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Center at (0, 0) The standard form of the equation of a circle with its center at the origin and radius r is

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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)

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(2) General equation of a circle is x 2 + y 2 + Dx + Ey + F = 0 circle coefficientsx 2 y 2 equal Note: The given quadratic relation will be a circle if the coefficients of the x 2 term and y 2 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 x 2 + y 2 + Dx + Ey + F = 0 at the point P(x 1,y 1 ) exactly one tangent In other words, a line that intersects the circle in exactly one point is said to be tangent to the circle..

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The equations of the two tangents with slope m to the circle x 2 + y 2 = r 2 are slope = m

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Condition for tangency A straight line y = mx + c is a tangent to the circle x 2 + y 2 = r 2 if and only if c 2 = r 2 ( 1 + m 2 ).

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Length of the tangent from the point P(x 1,y 1 ) to the circle x 2 + y 2 + Dx + Ey + F = 0 is (x 1,y 1 )

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Common chord b\w two circles. Common chord / tangent of two circles x 2 + y 2 + D 1 x + E 1 y + F 1 = 0 and x 2 + y 2 + D 2 x + E 2 y + F 2 = 0 is given by: (D 1 - D 2 )x + (E 1 - E 2 )y + (F 1 - F 2 )=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 x 2 + y 2 + Dx + Ey + F = 0 at the point P(x 1, y 1 ) is P(x 1,y 1 )

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Circles passing through the intersection of the circle x 2 + y 2 + Dx + Ey + F = 0 and the straight line Ax + By + C =0 Family of circles passing through the intersection of the two circles x 2 + y 2 + D 1 x + E 1 y + F 1 = 0 and x 2 + y 2 + D 2 x + E 2 y + F 2 = 0 x 2 + y 2 + Dx + Ey + F +k(Ax + By + C)= 0 x 2 + y 2 + D 1 x + E 1 y + F 1 + k(x 2 + y 2 + D 2 x + E 2 y + F 2 )= 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(x 1, y 1 ) be a point lying outside the circle x 2 + y 2 + Dx + Ey + F = 0. Then the equation

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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

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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 byradiuscenter The Cartesian equation of a sphere centered at the point (x 0,y 0,z 0 ) 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.spheresplane

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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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