Question

Derive an equation to the straight line in normal form p=x cos α + y sin α​

Answer 1

$\large\underline{\sf{Solution-}}$

Let us consider a line which makes an intercept of a units and b units on x - axis and y - axis respectively and is such that at a distance of p units from the origin and makes an angle α with the positive direction of x - axis.

We know, Equation of line which makes an intercept of a units and b units on x - axis and y - axis respectively is given by

$\rm \: \dfrac{x}{a} + \dfrac{y}{b} = 1 - - - - (1)$

Now, in right triangle OAP

$\rm \: cos \alpha = \dfrac{OP}{OA}$

$\rm \: cos \alpha = \dfrac{p}{a}$

$\rm \: a = \dfrac{p}{cos \alpha }$

$\rm\implies \:a \: = \: p \: sec \alpha - - - (2)$

Now, In right triangle OPB

$\rm \: cos(90 \degree - \alpha ) = \dfrac{OP}{OB}$

$\rm \: sin \alpha = \dfrac{p}{b}$

$\rm \: b = \dfrac{p}{sin \alpha }$

$\rm\implies \:\rm \: b = p \: cosec \alpha - - - (3)$

On substituting the values of a and b in equation (1), we get

$\rm \: \dfrac{x}{p \: sec \alpha } + \dfrac{y}{p \: cosec \alpha } = 1$

$\rm \: \dfrac{x}{ \: sec \alpha } + \dfrac{y}{ \: cosec \alpha } = p$

$\rm\implies \:x \: cos \alpha + y \: sin \alpha = p$

Hence, Proved

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Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to y - axis passes through the point (a, b) is x = a.

Equation of line parallel to x - axis passes through the point (a, b) is y = b.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.