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Slope intercept form

The equation of a line with slope m and making an intercept c on the y-axis is y = mx + c.

Point slope form

The equation of a line which passes through the point (x1, y1) and has the slope “m” is yy1 = m(xx1).

Two point form

The equation of a line passing through two points (x1, y1) and (x2, y2) is
yy1 = 73245.png(xx1)

Intercept form

The equation of a line which cuts off intercepts a and b respectively from the x- and y-axes is 73239.png.
 
74033.png

Normal form

The equation of the straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle α with the x-axis is x cos α + y sin α = p.

Equation of a line parallel to a given line

The equation of a line parallel to a given line ax + by + c = 0 is ax + by + λ = 0, where λ is a constant.

 

Note: The value of λ can be determined by some given conditions.

Equation of a line perpendicular to a given line

The equation of a line perpendicular to a given line ax + by + c = 0 is bxay + λ = 0, where λ is a constant.

Equations of straight lines through (x1, y1) making ∠α with y = mx + c

yy1 = tan(θ ± α) (xx1), where m = tan θ

Distance form of a line (parametric form)

The equation of the straight line passing through (x1, y1) and making an angle θ with the positive direction of the x-axis is
 
73179.png
 
where r is the distance of the point (x, y) on the line from point (x1, y1).

 

Notes:
  • The equation of the line is 73769.png = r
⇒ x – x1 = r cos θ and y – y1 = r sin θ
⇒ x = x1 + r cos θ and y = y1 = r sin θ
 
Thus, the coordinates of any point on the line at a distance r from the given point (x1y1) are (x1 + r cos θy1 + r sin θ). If P is on the right side of (x1y1), then r is positive and if P is on the left side of (x1y1), then r is negative. Since different values of r determine different points on the line, therefore the above form of the line is also called parametric form or symmetric form of a line.
  • At a given distance r from the point (x1y1) on the line 74081.png, there are two points, viz.
     
    (x1 + r cos θy1 + r sin θ) and (x1 – r cos θy1 – r sin θ)




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