Distance between two points.

The distance between two points

and

is given by

The distance of a point (x,y) from the origin is given by

Concurrent lines
If the lines

are concurrent, then

Angle between two lines

Let the two lines

make angles

with the positive x axis. Then,

.

The angle between the lines is given by

For parallel lines,

and for perpendicular lines,

The angle between the lines

is given by

If the lines are parallel,

.

If the lines are perpendicular to each other,

Perpendicular distance from origin
The perpendicular distance from the origin to the line ax+by+c=0 is

Perpendicular distance of a line from a point

The perpendicular distance from the point

to the line ax+by+c=0 is given by

Circle through three points

Given three points

the equation of the circle passing through the three points is

Conics

A conic is defined as the locus of a point which moves such that its distance from a fixed point is always e times its distance from a fixed straight line. The fixed point is called the focus of the conic and the fixed straight line is called the directrix of the conic. The constant ratio, e, is called the eccentricity of the conic.
If e<1, the conic is a parabola. If e=1, the conic is an ellipse and if e>1, the conic is a hyperbola.

Parabola

Standard form of a parabola:-

The coordinates of the focus are (a,0) and the vertex of the parabola is (0,0), the curve is symmetric about the x-axis.

Equation of the tangent at (x1, y1)

Condition that y=mx+c may be a tangent:-

The point of contact is

Latus rectum = 4a

Eccentricity, e=1

Equation of directrix, x=-e.

Equation of normal at (x1, y1):-

Equation of chord of contact of tangents drawn from (x1,y1):-

Parametric Representation

The coordinates (at², 2at) satisfies the equation y²=4ax.

The equation of the chord joining the points t1 and t2 on the parabola y²=4ax is

The equation of the tangent at (at², 2at) is

The equation of the normal at (at², 2at) is

The point of intersection of tangents at t1 and t2 is

Ellipse

Standrad form:-

The coordinates of the foci are (ae, 0) and (-ae, 0).
Tangent at (x1,y1)

Condition that y=mx+c may be a tangent:-

Latus rectum :

Eccentricity,

Equation of directrices:-

Equation of normal at (x1, y1):-

Equation of chord of contact of tangents drawn from (x1, y1):

Parametric Representation

x=acosθ, y=bsinθ satisifies the equation of the ellipse.

The equation of the tangent at θ is

The equation of normal at θ is

Hyperbola

Standard form:-

Tangent at (x1, y1):-

Condition that y=mx+c may be a tangent:-

Latus rectum :-

Eccentricity, e

Equation of directrices:-

Equation of normal at (x1, y1):-

Equation of chord of contact of tangents drawn from (x1,y1):-

An asymptote of a hyperbola is a straight line which touches the hyperbola at infinity but does not lie altogether at infinity.

The equations of asymptotes of a hyperbola are

Parametric Representation

x=asecθ, y=btanθ satisifies the equation of the hyperbola.

The equation of the tangent at θ is

The equation of the normal at θ is

Rectangular Hyperbola

A hyperbola in which b=a is called a Rectangular hyperbola. The asymptotes of a rectangular hyperbola are at right angles to each other.
The asymptotes of a Rectangular Hyperbola are

Standard form:-

Tangent at (x1, y1):-

Latus rectum = 2a

Eccentricity, e= √2

Equation of normal at (x1, y1):-

Equation of chord of contact of tangents drawn from (x1, y1):-

Parametric form

The point

satisifes the equation of the rectangular hyperbola,

The equation of the tangent at t is

The equation of the normal at t is

Angle between two planes

Let the equations of two planes be
ax+by+cz+d=0 and
a1x+b1y+c1z+d1=0

The angle between the two planes is defined as the angle between the normals.

If θ is the angle between the normals, then

Translation of Axes

The coordinates (x', y') of a coordinate system with origin O' = (x₀, y₀) relative to the standard origin (0, 0) are related to the standard (x, y) coordinates by the transformation equations

or

Rotation of Axes

The coordinates (x', y') of a coordinate system with origin (0, 0) with the x'-axis making an angle of α with the positive x-axis are related to the standard (x, y) coordinates by the transformation equations

or

Translation and Rotation of Axes

The coordinates (x', y') of a coordinate system with origin O' = (x₀, y₀) relative to the standard origin (0, 0) and with the x'-axis making an angle of α with the positive x-axis are related to the standard (x, y) coordinates by the transformation equations

or

Turning point of a parabola of the form y=ax² +bx+c

    or...   

Last edited by Identity (2007-08-12 21:15:59)