### 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

గమనిక : ప్రతిభ.ఈనాడు.నెట్లో వచ్చే ప్రకటనలు అనేక దేశాల నుండి, వ్యాపారస్తులు లేదా వ్యక్తుల నుండి వివిధ పద్ధతులలో సేకరించబడతాయి. ఆయా ప్రకటనకర్తల ఉత్పత్తులు లేదా సేవల గురించి ఈనాడు యాజమాన్యానికీ, ఉద్యోగస్తులకూ ఎటువంటి అవగాహనా ఉండదు. కొన్ని ప్రకటనలు పాఠకుల అభిరుచిననుసరించి కృత్రిమ మేధస్సు సాంకేతికతతో పంపబడతాయి. ఏ ప్రకటనని అయినా పాఠకులు తగినంత జాగ్రత్త వహించి, ఉత్పత్తులు లేదా సేవల గురించి తగిన విచారణ చేసి, తగిన జాగ్రత్తలు తీసుకొని కొనుగోలు చేయాలి. ఉత్పత్తులు / సేవలపై ఈనాడు యాజమాన్యానికి ఎటువంటి నియంత్రణ ఉండదు. కనుక ఉత్పత్తులు లేదా సేవల నాణ్యత లేదా లోపాల విషయంలో ఈనాడు యాజమాన్యం ఎటువంటి బాధ్యత వహించదు. ఈ విషయంలో ఎటువంటి ఉత్తర ప్రత్యుత్తరాలకీ తావు లేదు. ఫిర్యాదులు తీసుకోబడవు.