Normal Form

This lesson will cover another form of the equation of a straight line, called the normal form.

This form is less frequently used as compared to the previous ones. You wouldn’t see it around much. But nevertheless, we must have a look at what it’s like.

In this case, the length of the perpendicular from origin to the line is given to be p, and this perpendicular makes an angle α with the X-axis.

Confused? Here’s what I mean..

Straight Line Normal Form Derivation

Here, the perpendicular distance of the line OP equals p, and OP makes an angle α (measured anticlockwise) with the X-axis.

Once again, I’ll illustrate two different methods to derive the equation, one of which will use a previous form, and the other from scratch.

Method I

Les constructions..

Straight Line Normal Form Derivation

P is any point on the line. PF is perpendicular to the X-axis, FD perpendicular to OC, and PE perpendicular to FD. Quite complicated !

Not quite. Firstly, OC = p (given), which is OD + DC. Next, OD = OF cosα = xcosα. Finally, CD = PE = PF sinα = ysinα (angle PFE = α).

From the first relation, we get OD + DC = p. From the next two, after substituting the values of OD and DC, we get xcosα + ysinα = p

We’re done !

The required equation is xcosα + ysinα = p. (That was quick.)

Remember what we did (and what we’ll always do) when deriving the equation. Take any point P(x,y) on the line and establish a relation between x and y (and the given constants), which will always hold true.

Another method..

Method II

This time, I’ll use the intercept form of the line (this one) to derive the normal form of the equation. Have a look at the figure below.

Straight Line Normal Form Derivation

Now, what I’ve done is, expressed the intercepts in terms of the given information (p, α).

Since we now know the intercepts, we can use the intercept form of the equation, i.e. \(\frac{x}{psecα}+\frac{y}{pcosecα}=1\)

And, we get the same equation back, which is xcosα + ysinα = p

Lesson Summary

  1.  The equation of the line, whose perpendicular distance from origin is p, and this perpendicular makes an angle α with the X-axis, is given by xcosα + ysinα = p.

That’s it for now. See you in the next lesson with some examples.

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