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Straight Line

Distance Formula

If two point A (x₁ , y₁ ) and B(x₂ , y₂) are given , The distance between them is

Section Formula

Consider two points A(x₁ , y₁ ) and B(x₂ , y₂)
Case 1 : Let a point P(x,y) divides the segment AB internally in the ratio of m : n such that PA : PB = m : n

The coordinates of point P(x,y) :

Consider two points A(x₁ , y₁ ) and B(x₂ , y₂)
Case 2 : Let a point P(x,y) divides the segment AB externally in the ratio of m : n such that PA : PB = m : n

The coordinate of P (x,y) are :

Mid point Formula

If P(x,y) is the mid point of AB , then m:n = 1:1 , then the coordinate of point P :

Area of triangle

Consider a triangle with vertices A(x₁,y₁) , B(x₂,y₂) , C(x₃,y₃)

Slope of a Line

Slope of a line is defined as the tangent of tha angle θ which a line makes with +ve X-axis. It is denoted by m.
m= tanθ

Important Points :
i) m can be defined as tanθ for 0 ≤ θ ≤ π and θ ≠ π/2
ii) The Slope ofa line parallel to X-axis = 0 and perpendicular to x-axis is undefined.

Slope of a Line Using two points

Let the two points be A(x₁ , y₁) and B(x₂ , y₂)
Slope of line m = tanθ

Intercepts of a line :

The line L cuts X and y axes such that distance between the points where line cuts axes and origin is known as intercept.

X intercept = OA
Y-intercept = OB
Note : i) If Line is parallel to x-axis , X-intercept is undefined
ii) If line is perpendicular to X-axis , y intercept is undefined.

Different forms of equation of straight line :

1. Equation of a line L of slope 'm' and y intercept 'b'

y = mx + b where m is the slope and b is y intercept
this form is also known as slope-intercept form.

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2. Equation of a line L of slope 'm' and passing through a given point (x₁ , y₁) :

y-y₁ = m(x-x₁) where m is the slope
this form is also known as point-slope form.

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3. Equation of a line L passing through two given point (x₁ , y₁) and (x₂ , y₂) :

this form is also known as two-point form.

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4. Equation of line L cutting intercept 'a' and 'b' on X and Y axes :

where a and b are x intercept and y intercept respectively
this form of equation is known as intercept form.

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5. Equation of Line L in terms of p (the length of perpendicular from origin on Line) and α( the angle which p makes with +ve axis.)

Equation :

xcosα + ysinα = p

This equation is known as normal form of equation of line.

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6. General equation of straight line L :

Ax + By + C = 0

i) Slope : m = -A/B
ii) X-intercept : a = -C/A
iii) Y-intercept : b = -C/B

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Angle Between two lines.

Consider the slope of first Line L₁ = m₁ and the slope of first Line L₂ = m₂
If α is the acute angle between L₁ and L₂

i) If L₁ is parallel to L₂ ⇒ m₁ = m₂
ii) If L₁ is prependicular to L₂ ⇒ m₁.m₂ = -1
iii) Obtuse angle between the lines = π - α

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Perpendicular distance from a point to line.

a) Distance(d) of a point P(x₁ , y₂) from the line L : Ax + By + C = 0 :

b) Distance of Line : Ax+ By + C = 0 from origin (0 , 0) :

c) Distance between two parallel lines Ax+By + C₁ =0 and Distance between two parallel lines Ax+By + C₂ =0 is

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Concurrency of three lines

The lines A₁x+B₁y + C₁ = 0 , A₂x+B₂y + C₂ = 0 and A₃x+B₃y + C₃ = 0 pass through a common point if :

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Lecture for straight line part 1 :

Lecture for straight line part 2 :

Lecture for straight line part 3 :

Lecture for straight line part 4 :

Lecture for straight line part 5 :

Lecture for straight line part 6 :

Lecture for straight line part 7 :

Practice set 1 Practice set 2 Practice set 3