Title XI - maths - chapter 11 - PARABOLA (60-80).pdf ✅

60 NARAYANAGROUP PARABOLA JEE-MAIN SR-MATHS VOL-IV PARABOLA  Definition of a Conic : The locus of a point which moves in a plane such that the ratio of its distance from a fixed point to its perpendicular distance from a fixed straight line is always a constant, is called a conic section or a conic. M P(x, y) S( , )   Focus Directrix ax + by + c = 0 Let S be a fixed point and L be a fixed line, P be any point in the plane. Let M be the projection of P on L. Then the locus of P such that SP e PM  constant, is called a conic. The fixed point S is called the focus, the fixed line L is called the directrix and the constant ratio is denoted by e, is called the eccentricity of the conic. If e = 1, the conic is a parabola If 0 < e < 1, the conic is an ellipse If e > 1, the conic is a hyperbola If e = 0, the conic is a circle If e  , the conic is a pair of straight lines.  General Equation of a Conic Equation of a conic with  x y 1 1 , focus lx my n    0 directrix and eccentricity e is         2 2 2 2 2 2 1 1 l m x x y y e lx my n            .  The general equation of a conic is of the form 2 2 ax hxy by gx fy c       2 2 2 0 .  Recognisation of Conics : The equation of a conic is represented by the following general equation of second degree : 2 2 ax hxy by gx fy c       2 2 2 0 ...1 Case 1 :When 2 2 2        abc fgh af bg ch 2 0 Equation (1) represents a pair of straight lines. Condition Conic i) 2    0 & h ab a pair of parallel lines ii) 2    0 & h ab a pair of intersecting lines iii) 2    0 & h ab a pair of imaginary lines intersecting at a real point Case 2: If   0 Condition Conic i)     0, 0 , 0 a b h a circle ii) 2    0,h ab a parabola iii) 2    0,h ab an ellipse (or an empty set) iv) 2    0,h ab a hyperbola v) 2      0, , 0 h ab a b a rectangularhyperbola Definitions : 1. Principal Axis (Axis):The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section. 2. Vertex (Vertices):The point(s) of intersection of the conic section and the axis is (are) called the vertex (vertices) of the conic. 3. Chord :The line segment joining any two points on the conic is called a chord. 4. Focal chord : Any chord passing through the focus is called a focal chord of the conic section. 5. Double ordinate : A chord passing through a point P on the conic and perpendicular to the axis of the conic is called the double ordinate of the point P. 6. Latus rectum : The double ordinate passing through the focus is called the latus rectum of the conic. 7. Centre : The point which bisects every chord of the conic passing through it is called the centre of the conic. If the equation of conic is 2 2 ax hxy by gx fy c       2 2 2 0 , then the coordinates of centre of conic is 2 2 2 , ( ) hf bg gh af ab h ab h ab h            . SYNOPSIS