Analytical Geometry

Equation of Straight lines

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General Equation of a LineGeneral equation of first degree in two variables,

Ax + By + C= 0 where A, B and C are real constants such that A and B are not zero simultaneously.

Graph of the equation Ax + By + C= 0 is always a straight line.

Therefore, any equation of the form Ax + By + C= 0, where A and B are not zero simultaneously is called general linear equation or general equation of a line.

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Different forms of Ax + By + C = 0The general equation of a line can be reduced into

various forms of the equation of a line, by the following procedures:

Slope-intercept formIf B ≠ 0, then Ax + By + C= 0 can be written as

Where and then from equation (1) the slope of the st. line is and y intercept

is If B= 0 then x = which is a vertical line whose

slope is undefined and x-intercept is

B

Cx

B

Ay or y = mx + c -------------

(1)

B

Am

B

Cc

B

A

A

C

B

C

A

C

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Intercept form If C ≠0, then Ax + By + C = 0 can be written as

where We know that equation (1) is intercept form of the equation of a line whose

x-intercept is y- intercept is

If C = 0, then Ax + By + C = 0 can be written as Ax + By = 0, which is a line passing through the origin

and, therefore, has zero intercepts on the axes.

1

BCy

ACx

or 1b

y

a

x

A

Ca

B

Cb An

d

A

C

B

C

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Normal formLet x cos θ +y sin θ = p be the normal form of the line represented by the equation

Ax + By + C = 0 or Ax + By = – C. Thus, both the equations are same and therefore

Which gives, and

Now,

Or

p

CBA

sincos

C

Apcos

C

Bpsin

1sincos2

22

C

Bp

C

Ap

22

22

BA

Cp

or

22 BA

Cp

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Therefore and

Thus, the normal form of the equation Ax + By + C = 0 is

x cos θ + y sin θ = p, where

,

and

Proper choice of signs is made so that p should be

positive.

22cos

BA

A

22sin

BA

B

22cos

BA

A

22

sinBA

B

22 BA

Cp

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Distance of a Point From a Line

The distance of a point from a line is the length of the perpendicular drawn from the point to the line.Let L : Ax + By + C = 0 be a line, whose distance from the point P (x1, y1) is d.

Draw a perpendicular PM from the point P to the line L.

If the line meets the x-and y-axes at the points Q and R, respectively.

Then, coordinates of the points are

L : Ax + By + C = 0

X

Y

P(x1, y1)

B

CR ,0

0,A

CQ

B

CR ,0

0,A

CQ and

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Thus, the area of the triangle PQ is given byarea ( PQR)

, which givesAlso,

area ( PQR)

Or 2 area (ΔPQR)

QRPM .2

1

QR

PQR) (triangle area2PM

0002

1111

yy

B

C

A

C

B

Cx

AB

C

A

Cy

B

Cx

2

112

1

,. 11 CBxAxAB

C

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2222

00 BAAB

C

B

C

A

CQR

Substituting the values of area (ΔPQR) and QR in (1), we get

or

Thus, the perpendicular distance (d) of a line Ax + By+ C = 0 from a point (x1, y1) is given by

22

11

BA

CByAxPM

22

11

BA

CByAxd

22

11

BA

CByAxd

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Distance between two parallel linesWe know that slopes of two parallel lines are equal.Therefore,

two parallel lines can be taken in the formy = mx + c1 ... (1)

and y = mx + c2 ... (2)

Line (1) will intersect x-axis at the point

Distance between two lines is equal to the length of the perpendicular from point A to line (2).

X

Y

y = m

x + c 1

y = m

x + c 2

d

o

0,1

m

cA

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Therefore, distance between the lines (1) and (2) is

orThus,

the distance d between two parallel lines y = mx + c1 and y = mx + c2 is given by

If lines are given in general form, i.e., Ax + By + C1 = 0 and Ax + By + C2 = 0,

then above formula will take the form

21 cm

cm

2

21

1 m

ccd

2

21

1 m

ccd

22

21

BA

ccd

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