### Straight Lines

DEFINITION: A straight line is defined as the curve which is such that the line segment joining any two points lies on it.
Every first degree equation in x, y represents a straight line equation.
i.e. ax + by + c = 0 (a2 + b2 ≠ 0) represents a straight line.
If a line makes an angle 'θ' with the positive direction of the X − axis in anti clock wise sense then 'θ' is called inclination of a line. Where θ  [0 Π) and 'Tanθ' is defined as slope of the line. The slope of the line is denoted by 'm' (m = Tanθ)
Since a line parallel to X − axis makes an angle of '0º' with the X − axis then its slope m = Tan0º = 0.
Since a line parallel to Y − axis makes an angle of 90º with the X − axis then its slope m = Tan 90º = undefined (∞).
Since if two lines parallel they have same inclination then their slopes are equal.
If a line passing through two points A (x1, y1) and B(x2, y2) then its slope is
If the 'θ' is acute angle between two lines having slopes 'm1' and 'm2' then

If two lines parallel then θ = 0   m− m2 = 0  m1 = m2
If two lines perpendicular then θ =    1 + m1 m2 = 0
m1m2 = −1

DIFFERENT FORMS OF THE EQUATION OF A LINE

1) Line parallel to X − axis
If a straight line parallel to X − axis and is at a distance of 'a' units from it an above the X − axis then its equation is y = a

If a straight line parallel to Y - axis and at a distance of 'a' units from it below the X - axis then its equation is y = −a.

(2) Line parallel to Y - axis
If a straight line parallel to Y − axis and at a distance of 'b' units from it on right side of Y − axis then its equation is x = b

If a straight line parallel to Y − axis and at a distance of 'b' units from it on left side of Y − axis then its equation is x = −b

(3) Slope intercept form

The equation of a line with slope 'm' and making an intercept 'c' on Y − axis is y = mx + c

(4) point slope form
The equation of a line passing through (x1, y1) and having slope 'm' is y − y1 = m(x − x1)

(5) Two point form of a line
The equation of a line passing through (x1, y1) and (x2, y2) is

(6) The intercept form of a line
The equation of the straight line which cuts off intercepts of lengths of 'a' and 'b' on X − axis and Y − axis respectively is
Where 'a' is called x − intercept and 'b' is called y − intercept.
Where the line cuts X − axis at (a, 0) and Y − axis at (0, b)

(7) Normal form of a line
The equation of straight line which is at a distance of 'p' units from the origin   and it's normal (OP) makes an angle 'α' (0 α 2Π)  with positive direction of X − axis is x cosα + y sin α = p where p > 0 and 0 α

REDUCTION OF GENERAL EQUATION TO STANDARD FORM

General equation of straight line Ax + By + C = 0 (A2 + B2 ≠ 0)

(1) Reduction to 'Slope-Intercept' form
If B≠0 then Ax + By + C = 0 can be written as

Compare (i) with y = mx+c then slope m =
Y intercept c =

(2) Reduction to 'Intercept' form
If C ≠ 0 then Ax + By + C = 0 can be written as

Compare (ii) with    then
x − intercept a =  ; y − intercept b =

(3) Reduction to 'Normal' form
Let Ax + By + C = 0 can be written as Ax + By = −C ................ (iii)
compare (iii ) with xcosα + ysinα = P then we get

Here we have to choose the signs of cosα, sinα so that 'P' should be + ve

PARAMETRIC FORM OF A LINE

The equation of a straight line passing through (x1, y1) and having inclination θ is

= r.
Where 'r' is distance of the point (x, y) on the line from the point (x1, y1).
The co-ordinates of the points on the line
which are at a distance of 'r' units from the point (x1, y1) is (x1 ± r cosθ, y1 ± rsinθ)

Equation of a line which is parallel to the given line ax + by + c = 0 is ax + by + k = 0 (k is some constant)
Equation of a line which is perpendicular to the given line ax + by + c = 0 is bx − ay + λ = 0 (λ is some constant).

POSITION OF TWO POINTS RELATIVE TO A LINE

The points (x1, y1) and (x2, y2) lie on the same side or opposite sides of the given line ax + by + c = 0 according as (ax1 + by1 + c) (ax2 + by2 + c) > 0 or < 0 respectively.
The point (x1, y1) will lies on origin side of the line ax + by + c = 0 if (ax+ by+ c) c > 0
The point (x1, y1) will lies on non-origin side of the line ax + by + c = 0 if (ax+ by+ c) c < 0

POINT OF INTERSECTION OF TWO LINES

Point of intersection of the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is

Any line through the point of intersection of the lines a1x + b1y + c= 0 and a2x + b2y + c2 = 0 is of the form a1x + b1y + c1 + λ (a2x + b2y + c2) = 0 where λ  R

DISTANCE FROM A POINT TO A LINE

The distance (perpendicular distance) from a point (x1, y1) to a line ax + by + c = 0 is

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

The distance between two parallel lines ax + by + c= 0 and ax + by + c2 = 0 is

EQUATIONS OF STRAIGHT LINES PASSING THROUGH A GIVEN POINT AND MAKING A GIVEN ANGLE WITH A GIVEN LINE

Equation of straight lines passing through the given point (x1, y1) and making given angle 'α' with the given line y = mx + c is y − y1 = Tan (θ ± α) (x − x1) where Tan θ = m

EQUATIONS OF THE BISECTORS

The equations of the bisectors of the angles between the lines
a1x + b1y + c1 = 0 ........... (1) and
a2x + b2y + c2 = 0 ........... (2) is

EQUATION OF THE BISECTORS OF THE ANGLE CONTAINING THE ORIGIN AND NOT CONTAINING THE ORIGIN

Equation of the bisector of the angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 (where c1 > 0 and c2 > 0) containing the origin is

Equation of the bisector of the angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 (where c> 0 and c> 0) not containing the origin is

.
The equation of the bisector of the angle between the two lines a1x + b1y + c1 = 0 and a1x + b2y + c2 = 0 containing the point (h, k) will be

.

a1h + b1k + c1 and a2h + b2k + c2 are of the same sign or opposite sign
Let the equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 where c1 > 0; c2 > 0 and a1b2 ≠ a2b1
the equations bisectors are

(i) If (a1a2 + b1b2) > 0 then '+' sign gives obtuse angular bisector and '−' sign gives acute angular bisector.
(ii) (a1a2 + b1b2) < 0 then '+' sign gives acute angular bisector and '−' sign gives obtuse angular bisector.
Angular bisectors are perpendicular to each other.
If (a1a2 + b1b2) > 0 then origin lies in obtuse angle.
If (a1a2 + b1b2) < 0 then origin lies in acute angle.
The foot of the perpendicular drawn from the point (x1, y1) to the line ax + by + c = 0

If (h, k) is foot of the perpendicular drawn from the point (x1, y1) to the line  ax + by + c = 0 then
Image of a point (x1, y1) about a line ax + by + c = 0

If (h, k) is the image of a point (x1, y1) about the line ax + by + c = 0
then

The image of a point (α, β) with respect to X − axis is (α, − β)

The image of a point (α, β) with respect to Y − axis is (−α, β)

The image of a point (α, β) with respect to origin is (−α, −β)

The image of a point (α, β) with respect to the line x = a is (2a − α, β)

The image of a point (α, β) with respect to the line y = b is (α, 2b − β)

The image of a point (α, β) with respect to the line y = x is (β, α)

Posted Date : 17-02-2021

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గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.