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Conic Sections

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(1) Circle A circle is formed when i.e. when the plane is perpendicular to the axis of the cones.

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Conic Sections (2) Ellipse An ellipse is formed when i.e. when the plane cuts only one of the cones, but is neither perpendicular to the axis nor parallel to the a generator.

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Conic Sections (3) Parabola A parabola is formed when i.e. when the plane is parallel to a generator.

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Conic Sections (4) Hyperbola A hyperbola is formed when i.e. when the plane cuts both the cones, but does not pass through the common vertex.

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A circle is the locus of a variable point on a plane so that its distance (the radius)remains constant from a fixed point (the centre). y x O P(x,y)

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The standard equation of circle: where is the centre of the circle and r is its radius. × The parametric equation of a circle: × The general equation of a circle: where is the centre of the circle and is its radius

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Parabola A parabola is the locus of a variable point on a plane so that its distance from a fixed point (the focus) is equal to its distance from a fixed line (the directrix x = - a). focus F(a,0) P(x,y) M(-a,0)x y O

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Form the definition of parabola, PF = PN standard equation of a parabola

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mid-point of FM = the origin (O) = vertex length of the latus rectum =LL`= 4a vertex latus rectum (LL’) axis of symmetry

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Other forms of Parabola

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12.1 Equations of a Parabola A parabola is the locus of a variable point P which moves in a plane so that its distance from a fixed point F in the plane equals its distance from a fixed line l in the plane. The fixed point F is called the focus and the fixed line l is called the directrix.

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12.1 Equations of a Parabola The equation of a parabola with focus F(a,0) and directrix x + a =0, where a >0, is y 2 = 4ax.

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12.1 Equations of a Parabola X`X is the axis. O is the vertex. F is the focus. MN is the focal chord. HK is the latus rectum.

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The standard equation of parabola: where is the focus and is the vertex of parabola. × The parametric equation of a parabola: × The general equation of a parabola: with either a=0 or b=0 but both not zero at the same time.

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12.4 Equations of an Ellipse An ellipse is a curve which is the locus of a variable point which moves in a plane so that the sum of its distance from two fixed points remains a constant. The two fixed points are called foci. P’(x,y) P’’(x,y)

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Let PF 1 +PF 2 = 2a where a > 0

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standard equation of an ellipse

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vertex major axis = 2a minor axis = 2b lactus rectum length of semi-major axis = a length of the semi-minor axis = b length of lactus rectum = 12.4 Equations of an Ellipse

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AB major axis CD minor axis A, B, C and D vertices O centre PQ focal chord F focus RS, R’S’ latus rectum

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12.4 Equations of an Ellipse

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Other form of Ellipse where a 2 – b 2 = c 2 and a > b > 0 12.4 Equations of an Ellipse

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y x O (h, k)

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The standard equation of ellipse: where are the foci of the ellipse. × The parametric equation of an ellipse:

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A hyperbola is a curve which is the locus of a variable point which moves in a plane so that the difference of its distance from two points remains a constant. The two fixed points are called foci. P’(x,y) 12.7 Equations of a Hyperbola

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Let |PF 1 -PF 2 | = 2a where a > 0

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standard equation of a hyperbola

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vertex transverse axis conjugate axis lactus rectum length of lactus rectum = length of the semi-transverse axis = a length of the semi-conjugate axis = b

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12.7 Equations of a Hyperbola A 1, A 2 vertices A 1 A 2 transverse axis YY’ conjugate axis O centre GH focal chord CD lactus rectum

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asymptote equation of asymptote : 12.7 Equations of a Hyperbola

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Other form of Hyperbola : 12.7 Equations of a Hyperbola

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Rectangular Hyperbola If b = a, then The hyperbola is said to be rectangular hyperbola.

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equation of asymptote :

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12.7 Equations of a Hyperbola Properties of a hyperbola :

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12.7 Equations of a Hyperbola Parametric form of a hyperbola :

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12.8 Asymptotes of a Hyperbola

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Properties of asymptotes to a hyperbola :

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12.8 Asymptotes of a Hyperbola Properties of asymptotes to a hyperbola :

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Simple Parametric Equations and Locus Problems x = f(t) y = g(t) parametric equations parameter Combine the two parametric equations into one equation which is independent of t. Then sketch the locus of the equation.

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Equation of Tangents to Conics general equation of conics : Steps : (1) Differentiate the implicit equation to find. (2) Put the given contact point (x 1, y 1 ) into to find out the slope of tangent at that point. (3) Find the equation of the tangent at that point.

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Case I: If, the equation represents a circle with centre at and radius Case II: If and both have the same sign, the equation represents the standard equation of an ellipse in XY-coordinate system, where Case III: If and both have opposite signs, the equation represents the standard equation of hyperbola in XY-coordinate system, where Case IV: If,the equation represents the standard equation of parabola in XY- coordinate system, where

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With the understanding that occasional degenerate cases may arise, the quadratic curve is a parabola, if an ellipse, if a hyperbola, if

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In both ellipse and hyperbola, the eccentricity is the ratio of the distance between the foci to the distance between the vertices. Suppose the distance PF of a point P from a fixed point F (the focus)is a constant multiple of its distance from a fixed line (the directrix).i.e., where e is the constant of proportionality. Then the path traced by P is (a). a parabola if (b). an ellipse of eccentricity e if (c). a hyperbola of eccentricity e if

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ConicsParabolaEllipseHyperbola Graph Definitio n PF = PN PF 1 + PF 2 = 2a | PF 1 - PF 2 | = 2a

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ConicsParabolaEllipseHyperbola Graph Standard Equation

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ConicsParabolaEllipseHyperbola Graph Directrix x = -a

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ConicsParabolaEllipseHyperbola Graph Vertices (0,0) A(-a,0), B(a,0), C(0,b), D(0,-b) A 1 (a,0), A 2 (-a,0)

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ConicsParabolaEllipseHyperbola Graph Axes axis of parabola = the x-axis major axis = AB minor axis =CD transverse axis =A 1 A 2 conjugate axis =B 1 B 2 where B 1 (0,b), B 2 (0,-b)

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ConicsParabolaEllipseHyperbola Graph Length of lantus rectum LL ` 4a

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ConicsParabolaEllipseHyperbola Graph Asymptotes ----

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ConicsParabolaEllipseHyperbola Graph Parametric representation of P

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