We will learn how to find the equation of a straight line in 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 x-axis is x cos α + y sin α = p

If the line length of the perpendicular draw from the origin upon a line and the angle that the perpendicular makes with the positive direction of x-axis be given then to find the equation of the line.

Suppose the line AB intersects the x-axis at A and the y-axis at B. Now from the origin O draw OD perpendicular to AB.

The length of the perpendicular OD from the origin = p and ∠XOD = α, (0 ≤ α ≤ 2π).

Now we have to find the equation of the
straight line AB.

Now, from the right-angled ∆ODA we get,

\(\frac{OD}{OA}\) = cos α

⇒ \(\frac{p}{OA}\) = cos α

⇒ OA = \(\frac{p}{cos α}\)

Again, from the right-angled ∆ODB we get,

∠OBD = \(\frac{π}{2}\) - ∠BOD = ∠DOX = α

Therefore, \(\frac{OD}{OB}\) = sin α

or, \(\frac{p}{OB}\) = sin α

or, OB = \(\frac{p}{sin α}\)

Since the intercepts of the line AB on x-axis and y-axis are OA and OB respectively, hence the required

\(\frac{x}{OA}\) + \(\frac{y}{OB}\) = 1

⇒ \(\frac{x}{\frac{p}{cos α}}\) + \(\frac{y}{\frac{p}{sin α}}\) = 1

⇒ \(\frac{x cos α}{p}\) + \(\frac{y sin α}{p}\) = 1

⇒ x cos α + y sin α = p, which is the required form.

Solved examples to find the equation of a straight line in normal form:

Find the equation of the straight line which is at a of distance 7 units from the origin and the perpendicular from the origin to the line makes an angle 45° with the positive direction of x-axis.

**Solution:**

We know that 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 x-axis is x cos α + y sin α = p.

Here p = 7 and α = 45°

Therefore, the equation of the straight line in normal form is

x cos 45° + y sin 45° = 7

⇒ x ∙ \(\frac{1}{√2}\) + y ∙ \(\frac{1}{√2}\) = 7

⇒ \(\frac{x}{√2}\) + \(\frac{y}{√2}\) = 7

⇒ x + y = 7√2, which is the required equation.

**Note: **

(i) The equation of a, straight line in the form of x cos α + y sin α = p is called its normal form.

(ii) In equation x cos α + y sin α = p, the value of p is always positive and 0 ≤ α≤ 360°.

**●**** The Straight Line**

**Straight Line****Slope of a Straight Line****Slope of a Line through Two Given Points****Collinearity of Three Points****Equation of a Line Parallel to x-axis****Equation of a Line Parallel to y-axis****Slope-intercept Form****Point-slope Form****Straight line in Two-point Form****Straight Line in Intercept Form****Straight Line in Normal Form****General Form into Slope-intercept Form****General Form into Intercept Form****General Form into Normal Form****Point of Intersection of Two Lines****Concurrency of Three Lines****Angle between Two Straight Lines****Condition of Parallelism of Lines****Equation of a Line Parallel to a Line****Condition of Perpendicularity of Two Lines****Equation of a Line Perpendicular to a Line****Identical Straight Lines****Position of a Point Relative to a Line****Distance of a Point from a Straight Line****Equations of the Bisectors of the Angles between Two Straight Lines****Bisector of the Angle which Contains the Origin****Straight Line Formulae****Problems on Straight Lines****Word Problems on Straight Lines****Problems on Slope and Intercept**

**11 and 12 Grade Math**

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