Chapter-5-1.CONIC SECTION-01- (THEORY)

5.0.1. Introduction. The curves obtained by intersection of a plane and a double cone in different orientation are called conic section. In other words “Graph of a quadratic equation (in two variables) is a “Conic section”. A conic section or conic is the locus of a point P, which moves in such a way that its distance from a fixed point S always bears a constant ratio to its distance from a fixed straight line, all being in the same plane. constant = e (eccentricity) or SP= e. PM 5.0.2. Definitions of Various important Terms. (1) Focus : The fixed point is called the focus of the conic-section. (2) Directrix : The fixed straight line is called the directrix of the conic section. In general, every central conic has four foci, two of which are real and the other two are imaginary. Due to two real foci, every conic has two directrices corresponding to each real focus. (3) Eccentricity : The constant ratio is called the eccentricity of the conic section and is denoted by e. If , the conic is called Parabola. If e < 1, the conic is called Ellipse. If , the conic is called Hyperbola. If , the conic is called Circle. If , the conic is called Pair of the straight lines. (4) Axis: The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section. A conic is always symmetric about its axis. (5) Vertex: The points of intersection of the conic section and the axis are called vertices of conic section. (6) Centre: The point which bisects every chord of the conic passing through it, is called the centre of conic. (7) Latus-rectum: The latus-rectum of a conic is the chord passing through the focus and perpendicular to the axis. (8) Double ordinate: The double ordinate of a conic is a chord perpendicular to the axis. (9) Focal chord: A chord passing through the focus of the conic is called a focal chord. (10) Focal distance: The distance of any point on the conic from the focus is called the focal distance of the point. 5.0.3. General equation of a Conic section when its Focus, Directrix and Eccentricity are given Let be the focus, be the directrix and e be the eccentricity of a conic. Let be any point on the conic. Let PM be the perpendicular from P, on the directrix. Then by definition   Thus the locus of is this is the cartesian equation of the conic section which, when simplified, can be written in the form , which is general equation of second degree. 5.0.4. Recognisation of Conics. The equation of conics is represented by the general equation of second degree ......(i) and discriminant of above equation is represented by , where Case I: When In this case equation (i) represents the degenerate conic whose nature is given in the following table. S. No. Condition Nature of conic 1. and A pair of coincident straight lines 2. and A pair of intersecting straight lines 3. and A point Case II: When In this case equation (i) represents the non-degenerate conic whose nature is given in the following table. S. No. Condition Nature of conic 1. A circle 2. A parabola 3. An ellipse 4. A hyperbola 5. and A rectangular hyperbola 5.0.5. Method to find centre of a Conic. Let be the given conic. Find Solve for x, y we shall get the required centre (x, y) Example: 1 The equation represents (a) A parabola (b) An ellipse (c) A hyperbola (d) A circle Solution: (a) Comparing the given equation with Here, a = 1, b = 1, h = – 1, , f = 0, c = 2 Now i.e., and i.e., So given equation represents a parabola. Example: 2 The centre of is [BIT Ranchi1986] (a) (2, 3) (b) (2, –3) (c) (–2, 3) (d) (–2, –3) Solution: (a) Centre of conic is Here, , , , , Centre Centre .