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##### Unit 12

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**Unit 12**Conic Sections**A circle is formed when**i.e. when the plane is perpendicular to the axis of the cones. Conic Sections (1) Circle**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 (2) Ellipse**A parabola is formed when**i.e. when the plane is parallel to a generator. Conic Sections (3) Parabola**A hyperbola is formed when**i.e. when the plane cuts both the cones, but does not pass through the common vertex. Conic Sections (4) Hyperbola**y**P(x,y) M(-a,0) O focus F(a,0) x 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).**Form the definition of parabola,**PF = PN standard equation of a parabola**axis of symmetry**vertex latus rectum (LL’) mid-point of FM = the origin (O) = vertex length of the latus rectum = LL’= 4a**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 y2 = 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.**Equation of the tangent to the parabola y2 = 4ax at the**point (x1, y1) is y1y = 2a(x + x1), and that the point (at2, 2at) is 12.2 Chords and Tangents of a Parabola**12.2 Chords and Tangents of a Parabola**A straight line, not parallel to the axis of a parabola, is a tangent to the parabola if and only if it cuts the parabola at exactly one point.**12.2 Chords and Tangents of a Parabola**Remarks : In general, a tangent to a curve may not cut the curve at exactly one point. In parabola, the situation is quite different. A tangent to the parabola does not meet the curve in addition to the point of contact. This fact enables us to find the equation of the tangent algebraically. P Q**12.2 Chords and Tangents of a Parabola**Let P be a point not on the parabola such that two tangents, touching the curve at two points respectively, can be drawn from P to the parabola. The chord joining the points of contact is called the chord of contact of tangents drawn to the parabola from the point P.**12.2 Chords and Tangents of a Parabola**The equation of the chord of contact of tangents drawn to the parabola y2 = 4ax from the point P(x1, y1) is y1y = 2a(x + x1).**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)**length of lactus rectum =**12.4 Equations of an Ellipse major axis = 2a vertex lactus rectum minor axis = 2b length of semi-major axis = a length of the semi-minor axis = b**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 a2 – b2 = c2 and a > b > 0**12.4 Equations of an Ellipse**y (h, k) x O**12.5 Chords and Tangents of an Ellipse**A straight line is a tangent to an ellipse if and only if it cuts (touches) the ellipse at exactly one point.**12.7 Equations of a Hyperbola**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 it to two points remains a constant. The two fixed points are called foci. P’(x,y)**length of lactus rectum =**transverse axis vertex lactus rectum conjugate axis length of the semi-transverse axis = a length of the semi-conjugate axis = b**12.7 Equations of a Hyperbola**A1, A2 vertices A1A2 transverse axis YY’ conjugate axis O centre GH focal chord CD lactus rectum**12.7 Equations of a Hyperbola**asymptote equation of asymptote :**12.7 Equations of a Hyperbola**Other form of Hyperbola :**Rectangular Hyperbola**If b = a, then The hyperbola is said to be rectangular hyperbola.**12.7 Equations of a Hyperbola**Properties of a hyperbola :**12.7 Equations of a Hyperbola**Parametric form of a hyperbola :