Conic Sections

 Course Outlines

r     Parabola

·         Introduction of conic sections

·         Review of equations of tangent and normal

·         Equations of pair of tangents

·         Director circle

·         Chord of contact

·         Pole and polar

·         Properties of pole and polar

r     Ellipse

·         Equation of ellipse

·         Auxiliary circle and eccentric angles

·         Position of a point

·         Tangent and normal

·         Pair of tangents from an external   point

·         Director circle

·         Chord of contact

·         Pole and polar and their properties

·         Chord with a given middle point

·         Diameter, Conjugate diameter

·         Equi-conjugate diameters and its properties

 r    Hyperbola

·         Equation of hyperbola

·         Parametric coordinates and  conditions of tangency

·         Equation of tangent and normal

·         Chord or contact

·         Pair of tangents

·         Auxiliary circle and  director circle

·         Conjugate points and lines

·         Equation of chord

·         Rectangular hyperbola

·         Asymptotes and its equations

·         Equation of diameter and its properties

·         Conjugate diameters and its   properties

Introduction

    A conic section is the locus of a point which moves in a plane so that its distance from fixed point is in a constant ratio to its perpendicular distance from the fixed straight line. The fixed point is called focus and fixed straight line is called the directrix. The constant ratio is called eccentricity and it is denoted by e.

If e = 1, the conic is called parabola.

If e < 1, the conic is called an ellipse

If e > 1, the conic is called a hyperbola.

Solved Problems 

Q.1.      Define parabola and derive the equation of parabola in the standard form with vertex as the origin and axis is the x-axis.

Soln:    Parabola

 A parabola is the locus of a point which is equidistant from a fixed point (called the focus) and a fixed line (called the directrix).

To derive the equation of parabola in standard form with vertex at the origin and axis is the x-axis.

 Let S be focus and ZM, the directrix of the parabola. SZ is drawn perpendicular to ZM. Let A be the middle point of SZ, so that SA = AZ. Then A is the vertex and ZAS is the axis of the parabola.

 To determine the equation of a parabola in the simplest form (the standard form), take the vertex at the origin, the focus S on the x-axis so that the axis of the parabola is the x-axis and the directrix is parallel to the y-axis.

Let AS = a. Thus the coordinates of Z, A and S are respectively (– a, 0), (0, 0) and (a, 0) and the equation of directrix is x + a = 0.

Let P(x, y) be any point on the parabola. Join PS and draw PM perpendicular to ZM.

             Then,

                         PS = PM

             i.e.        PS² = PM²

             or,         (x – a)² + y² = (x + a)²

             or,         x² – 2ax + a² + y² = x² + 2ax + a²

             or,         y² = 4ax

             This is true for coordinates of any point on the parabola, hence it is the equation of parabola.