Analytic Geometry

Introduction:

Geometry is a study of points, lines, curves, surfaces, etc. and their properties. To bring a relationship between algebra and geometry Descartes introduced basic algebraic entity ‘number’ and basic geometric concept of ‘point’. This relationship is called ‘system of coordinates’.

Locus:

The path traced by a point when it moves according to specified geometrical conditions is called the locus of the point.

Example: The locus of point P(x₁, y₁) whose distance from a fixed point C (h, k) is constant ‘a’, is a circle.

Fixed point ‘C’ is called the centre and the fixed distance a’ is called the radius of the circle.

Straight line:

Straight is the simplest geometrical curve. The general form of the equation of straight line is Ax+By+C=0.

ax² + 2hxy +by² + 2gx + 2fy + c = 0 where a, b, c, f, g, h are constants.

Circle:

A circle is the locus of a point which moves in such a way that its distance from a fixed point is always constant.

The general equation of the circle is x² + y² + 2gx + 2fy + c =0 whose centre is (-g, -f) and radius is

Parabola:

The locus of a point whose distance from a fixed point is equal to its distance from a fixed line is called a parabola.

The general equation of the parabola is y² = 4ax with

Vertex (0, 0); Focus F(a, 0); Directrix x=-a; Latus rectum x=a.

Ellipse:

The locus of point in a plane whose distance from a fixed point bears a constant ratio, less than one to its distance from a fixed line is called ellipse.

The standard equation of the ellipse is with

Vertices:  A(a,o) and A’ (-a, 0)

Foci: (±ae,0)

Directrices:   x=

Latus rectum: x=(±ae

Eccentricity:

Hyperbola:

The locus of a point whose distance from a fixed point bears a constant ratio, greater than one to its distance from a fixed line is called a hyperbola.

The standard equation of the hyperbola is with

Vertices:  (a,o) and (-a, 0)

Foci: (±ae,0)

Directrices:   x=

Latus rectum: x=(±ae

Eccentricity: