Straight Lines

Straight Lines

Coordinate Geometry


Introduction

The most popular coordinate system is the rectangular Cartesian system. Here we consider the space to be two-dimensional. Through a point O, referred to as the origin, we take two

mutually perpendicular lines xOx' and yOy' and call them the x-axis and the y-axis respectively. x is called the x-coordinate or the abscissa of P and y is called the y-coordinate or the

ordinate of P.

The distance of the origin from the coordinate axes is zero, the coordinates of the origin are (0, 0).






Note :

1. If the position of point P (x, y) is known, then the lengths OM (= x) and MP (= y) can be determined.

2. If the lengths OM and MP are known, the coordinates of the point P are determined.

3. The distances measured along Ox and Oy are taken as positive, whereas, the distances measured along Ox' and Oy' are taken as negative.



Distance between two points

Let A and B be two given points, whose coordinates are given by A(x₁, y₁) and B(x₂, y₂) respectively. Then AB =.




Note:

Three given point A, B, C are collinear (or lie on a straight line) if the sum of the distances between two pairs of points is equal to the distance between the third pair of points.



The points A, B and C (arranged in that order) are collinear if AB + BC = AC.

Three non-collinear points always form a triangle.


Area of a triangle

Let (x₁, y₁), (x₂, y₂) and (x₃, y₃) respectively be the coordinates of the vertices A, B, C of a triangle ABC. Then the area of triangle ABC, is



= .



Area of a polygon of n sides

First of all we plot the points and see their actual order.

Let A₁(x₁, y₁), A₂(x₂, y₂),..., A_n(x_n, y_n) be the vertices of the polygon in anticlockwise order. Then area of the polygon

=.



Note:
If the vertices are in the clockwise order then take modulus.



Section formula
Coordinates of the point P dividing the join of two points A(x₁, y₁) and B(x₂, y₂) internally in the given ratio ₁ : ₂ are P.

1.Coordinates of the point P dividing the join of two points A(x₁, y₁) and B(x₂, y₂) externally in the ratio of₁ : ₂ are.

2. If the point P divides the line joining A and B internally in the ratio ₁ : ₂, it can also be said that P divides the line joining A and B externally in the ratio ₁ :- ₂.



Centroid of a triangle

The centroid of a triangle is the point of concurrency of its medians. The centroid G of the triangle ABC, divides the median AD, in the ratio of 2 : 1.

The centroid of the triangle, the coordinates of whose vertices are given by A(x₁, y₁), B(x₂, y₂) and

C(x₃, y₃) respectively, is .





Incentre of a triangle

The incentre 'I' of a triangle is the point of concurrency of the bisectors of the angles of the triangle.

The incentre of the triangle, the coordinates of whose vertices are given by A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), is

where a, b, c are the lengths of sides BC, CA, AB respectively.





Circum-centre of a triangle

This is the point of concurrency of the perpendicular bisectors of the sides of the triangle. This is also the centre of the circle, passing through the vertices of the given triangle.



Orthocentre of a triangle

This is the point of concurrency of the altitudes of the triangle.



Ex-Centre of a triangle

Excentre of a triangle is the point of concurrency of bisectors of two exterior and third interior angle. Hence there are three excentres I₁, I₂ and I₃ opposite to the three vertices of a

triangle.

If A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are the vertices of a triangle ABC,

co-ordinates of centre of ex-circle opposite to vertex A are given as

I₁(x, y) = .

Similarly co-ordinates of I₂(x , y) and I₃(x , y) are



I₂(x, y) = ,

I₃(x, y) = .



Locus

When a point moves in a plane under certain geometrical conditions, the point traces out a path. This path of the moving point is called its locus.



Equation of Locus

The equation to the locus is the relation which exists between the coordinates of all the points on the path, and which holds for no other points except those lying on the path.



Procedure for finding the equation of the locus of a point

(i). If we are finding the equation of the locus of a point P, assign coordinates (h, k) to P.

(ii) Express the given conditions as equations in terms of the known quantities and unknown parameters.

(iii). Eliminate the parameters, so that the eliminant contains only h, k and known quantities.

(iv). Replace h by x, and k by y, in the eliminant. The resulting equation is the equation of the locus of P.



Straight Line

Any equation of first degree of the form Ax + By + C = 0, where A, B, C are constants represents a straight line (at least one out of A and B is non zero).

1. The coordinates of any point on the x-axis are of the form (a, 0), where a may be positive or negative i.e. the ordinate of any point on the x-axis is zero. Hence the equation of the x-

axis is y = 0.

2. For a line parallel to the x-axis at a distance c from it, the ordinate of any point P on the line is c. Hence the equation of a line parallel to the x-axis is of the form y = c.

3. The coordinates of a point on the y-axis are (0, b) for positive or negative b so that the equation of the y-axis is x = 0.

4. For a line parallel to the y-axis at a distance d, the abscissa of any point on the line is d. Hence the equation of a line parallel to the y-axis is of the form x = d.



Slope

If is the angle at which a straight line is inclined to the positive direction of the x-axis, then m = tan , (0< 180^o) is called the slope of the line.

1.For a line parallel to the x-axis, = 0 tan = 0 so that the slope of the line parallel to the x-axis is zero.

2.For a line parallel to the y-axis, = /2 tan = so that the slope of the line parallel to the y-axis is (and hence not defined).

3.For 0 << /2, tan > 0 so that the slope of the line increases from zero to infinity (i.e. the line rises from right to left) as is increased from zero to /2.

4.For /2 << , tan < 0, the slope of the line decreases and the line falls from right to left as is increased from /2 to .

5. Since tan = tan ( + ), the slope of a line is independent of its direction.



Standard equations of the Straight Line

Slope Intercept From: y = mx + c, where

m = slope of the line

c = y intercept




The equation of the line is mx = y - c or y = mx + c.



Intercept Form: x/a + y/b = 1

x intercept = a

y intercept = b


The equation of the line AB is .



Normal Form:
x cosa + y sina = p, where a is the angle which the perpendicular to the line makes with the axis of x and p is the length of the perpendicular from the origin to the line. p is always

positive.



The equation of the line AB is

or x cos + y sin = p.




Slope point form

(a) One point on the straight line

(b) The gradient of the straight line i.e., the slope m of the line



Equation:
y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the straight line.

The equation of a line in the slope intercept form is

y = mx + c. Since the point (x₁, y₁) lies on it,

y₁ = mx₁ = c. Subtracting, we get

y - y₁ = m (x - x₁).




Two point form

Equation:



where (x₁, y₁) and (x₂, y₂) are the two given points. Here.



Note:

To fix the definite position of a straight line, two conditions must be given.

The general equation of a line i.e. ax + by + c = 0, involves three constants. However, dividing them by a non-zero constant (a or b or c), say c, we get

or a₁x + b₁y + 1 = 0.

Hence the general equation of a straight line contains two arbitrary constants. Hence, we can determine the equation of a particular line if two conditions are given.

Parametric form

To find the equation of a straight line which passes through a given point A(h, k) and makes a given angle , with the positive direction of the x-axis. P(x, y) is any point on the line

LAL'.

AP = r so that x - h = r cos ,

y - k = r sin

.

This is the parametric form of the equations of the straight line LAL'.

The coordinates of P are (h + r cos, k + r sin).



These are the coordinates of any point on the line at a distance r from the point A. For different values of r, we get different points on the line.



Angle between two straight lines
If q is the angle between two lines, then tan =

where m₁ and m₂ are the slopes of the two lines.

If tan (₁ - ₂) =.

Since angle can be acute or obtuse, we write tan = ,

and these values are supplementary to each other. Most often we are interested in the acute angle, so that

tan =.



Note:

(i). If the two lines are perpendicular to each other then m₁m₂ = -1.

Any line perpendicular to ax + by + c = 0 is of the form bx - ay + k = 0.

(ii). If the two lines are parallel or are coincident, then m₁ = m₂.

Any line parallel to ax + by + c = 0 is of the form ax + by + k = 0.





Length of the Perpendicular from a Point on a Line
The length of the perpendicular from P(x₁, y₁) on ax + by + c = 0 is.


The distance between two parallel lines

The distance between two parallel lines

ax + by + c₁ = 0 and ax + by + c₂ = 0 is.



Family of lines

The general equation of the family of lines through the point of intersection of two given lines is L + L' = 0, where L = 0 and L' = 0 are the given lines, andis a parameter. It can be

easily verified that the point (x₁, y₁) of intersection of the lines L = 0 and L' = 0 satisfies L + L' = 0 for all values of .



Note:
1. For an arbitrary constant k₁ the equation ax + by + k₁ = 0 represent a family of lines parallel to the given line ax + by + c = 0.

2. For an arbitrary constant k₂ the equation bx - ay + k₂ = 0 represents a family of lines perpendicular to the line ax + by + c = 0.



Concurrency of Straight Lines

Three lines are said to be concurrent if they meet at the same point.

The condition for 3 lines a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0, a₃x + b₃y + c₃ = 0 to be

concurrent is

(i) .


(ii) The three lines are concurrent if any one of the lines passes through the point of intersection of the other two lines.

We find the point of intersection of any two lines. If this point lies on the third line. i.e. its coordinates satisfy the equation of the third line, the three lines are concurrent.



Angle Bisectors

To find the equations of the bisectors of the angle between the lines

a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.

A bisector is the locus of a point, which moves such that the perpendiculars drawn from it to the two given lines, are equal.

The equations of the bisectors are



Pair of straight lines

The general equation of second degree ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a pair of straight lines

if abc + 2fgh - af² - bg² - ch² = 0 and h² ab.

The homogeneous second degree equation ax² + 2hxy + by² = 0 represents a pair of straight lines through the origin.

If lines through the origin whose joint equation is ax² + 2hxy + by² = 0, are y = m₁x and

y = m₂x, then y² - (m₁ + m₂)xy + m₁m₂x² = 0 and y² + xy + = 0 are identical. If q is the angle between the two lines,

then = .

The lines are perpendicular if a + b = 0 and coincident if h² = ab.



Translation of Axes:

It is sometimes desirable to simplify the equation of a given curve or to bring it to some standard from by shifting the origin of coordinates axes to some other point, the axes remaining

parallel to the original axes. This process is called the translation of axes.

when the origin is shifted from (0, 0) to (h, k), the axes remaining parallel to the original axes, we substitute x' + h for x and y' + k for y in the given equation and obtain the new one

which refers to the new system of coordinates