# Conic Sections. (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

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

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.

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

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.

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)

 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

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

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

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

Other forms of Parabola

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.

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.

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.

 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.

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)

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

standard equation of an ellipse

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

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

12.4 Equations of an Ellipse

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

y x O (h, k)

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

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

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

standard equation of a hyperbola

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

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

asymptote equation of asymptote : 12.7 Equations of a Hyperbola

Other form of Hyperbola : 12.7 Equations of a Hyperbola

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

equation of asymptote :

12.7 Equations of a Hyperbola Properties of a hyperbola :

12.7 Equations of a Hyperbola Parametric form of a hyperbola :

12.8 Asymptotes of a Hyperbola

Properties of asymptotes to a hyperbola :

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

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.

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.

 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

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

 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

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

ConicsParabolaEllipseHyperbola Graph Standard Equation

ConicsParabolaEllipseHyperbola Graph Directrix x = -a

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)

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)

ConicsParabolaEllipseHyperbola Graph Length of lantus rectum LL ` 4a

ConicsParabolaEllipseHyperbola Graph Asymptotes ----

ConicsParabolaEllipseHyperbola Graph Parametric representation of P

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