General Form into Normal Form

We will learn the transformation of general form into normal form.

To reduce the general equation Ax + By + C = 0 into normal form (x cos α + y sin α = p):

We have the general equation Ax + By + C = 0.

Let the normal form of the given equation ax + by + c = 0……………. (i) be  

x cos α + y sin α - p = 0, where p > 0. ……………. (ii)

Then, the equations (i) and (ii) are the same straight line i.e., identical.

⇒ $\frac{A}{cos α}$ = $\frac{B}{sin α}$ = $\frac{C}{-p}$

⇒ $\frac{C}{P}$ = $\frac{-A}{cos α}$ = $\frac{-B}{sin α}$ = $\frac{+\sqrt{a^{2} + b^{2}}}{\sqrt{cos^{2} α + sin^{2} α}}$ = +  $\sqrt{A^{2} + B^{2}}$

Therefore, p = $\frac{C}{\sqrt{A^{2} + B^{2}}}$, cos α = - $\frac{A}{\sqrt{A^{2} + B^{2}}}$ and sin α = - $\frac{B}{\sqrt{A^{2} + B^{2}}}$

So, putting the values of cos α, sin α and p in the equation (ii) we get the form,

⇒ - $\frac{A}{\sqrt{A^{2} + B^{2}}}$ x - $\frac{B}{\sqrt{A^{2} + B^{2}}}$ y - $\frac{C}{\sqrt{A^{2} + B^{2}}}$ =  0, when c > 0

⇒ $\frac{A}{\sqrt{A^{2} + B^{2}}}$ x +  $\frac{B}{\sqrt{A^{2} + B^{2}}}$ y = - $\frac{C}{\sqrt{A^{2} + B^{2}}}$, when c < 0

Which is the required normal form of the general form of equation Ax + By + C = 0.

Algorithm to Transform the General Equation to Normal Form

Step I: Transfer the constant term to the right hand side and make it positive.

Step II: Divide both sides by $\sqrt{(\textrm{Coefficient of x})^{2} + (\textrm{Coefficient of y})^{2}}$.

The obtained equation will be in the normal form.

Solved examples on transformation of general equation into normal form:

1. Reduce the line 4x + 3y - 19 = 0 to the normal form.

Solution:

The given equation is 4x + 3y - 19 = 0

First shift the constant term (-19) on the RHS and make it positive.

4x + 3y = 19 ………….. (i)

Now determine $\sqrt{(\textrm{Coefficient of x})^{2} + (\textrm{Coefficient of y})^{2}}$

= $\sqrt{(4)^{2} + (3)^{2}}$

= $\sqrt{16 + 9}$

= √25

= 5

Now dividing both sides of the equation (i) by 5, we get

$\frac{4}{5}$x + $\frac{3}{5}$y = $\frac{19}{5}$

Which is the normal form of the given equation 4x + 3y - 19 = 0.

2. Transform the equation 3x + 4y = 5√2 to normal form and find the perpendicular distance from the origin of the straight line; also find the angle that the perpendicular makes with the positive direction of the x-axis.

Solution:    

The given equation is 3x + 4y = 5√2 ……..….. (i)

Dividing both sides of equation (1) by + $\sqrt{(3)^{2} + (4)^{2}}$ = + 5 we get,

⇒ $\frac{3}{5}$x + $\frac{4}{5}$y = $\frac{5√2}{5}$

⇒ $\frac{3}{5}$x + $\frac{4}{5}$y = √2

Which is the normal form of the given equation 3x + 4y = 5√2.

Therefore, the required, perpendicular distance from the origin of the straight line (i) is √2 units.

If the perpendicular makes an angle α with the positive direction of the x-axis then,

cos α = $\frac{3}{4}$ and sin α = $\frac{4}{5}$

Therefore, tan α = $\frac{sin α}{cos α }$ = $\frac{\frac{4}{5}}{\frac{3}{5}}$ = $\frac{4}{3}$

⇒ α = tan$^{-1}$$\frac{4}{3}$.

● 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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