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Perpendicular Distance ( p )

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☼ Perpendicular Distance ( p ) :–

(i) If P (x₁, y₁) is a point outside a line ax + by + c = 0, then perpendicular distance from P to line is,

p = PM = $\frac{\left|ax+\mathrm{\ by\ }+\mathrm{\ c}\right|}{\sqrt{a^2\ +\mathrm{\ } \mathrm{b}^2}}$

(ii) If O (0, 0) is a point outside a line ax + by + c = 0, then perpendicular distance from O to line is,

p = OM = $\frac{\left|c\right|}{\sqrt{a^2\ +\mathrm{\ } \mathrm{b}^2}}$

(iii) The perpendicular distance between two parallel lines l₁ : a₁x + b₁y + c₁ = 0 and l₂ : a₂x + b₂y + c₂ = 0 is,

p = MN = $\frac{\left|c_1-c_2\right|}{\sqrt{a^2+b^2}}$

☼ The co–ordinates of two points at a distance |r| from the point P (x₁, y₁) are A (x₁ + |r|cosθ, y₁ + |r|sinθ) and B (x₁ – |r|cosθ, y₁ – |r|sinθ)

☼ Let A(x₁, y₁) and B (x₂, y₂). If line l : ax + by + c = 0 divides from A in the ratio λ, then

= – $\frac{\mathrm{a}\mathrm{x}_1\ +\mathrm{\ b} \mathrm{y}_1\ +\mathrm{\ c} }{\mathrm{a}\mathrm{x}_2\ +\mathrm{\ b} \mathrm{y}_2\ +\mathrm{\ c} }$