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# Different Forms Of The Equation Of Line

We know that there are infinite points in the coordinate plane. Consider an arbitrary point P(x,y)Â on the XYÂ plane and a line L. How will we confirm whether the point is lying on the line L? This is where the importance of equation of a straight line comes into the picture in two-dimensional geometry.

Equation of a straight line contains terms in xÂ and y. If the point P(x,y)Â satisfies the equation of the line, then the point PÂ lies on the line L.

## Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of the lines which are horizontal or parallel to the X-axisÂ is y = a,Â where a is the y â€“ coordinate of the points on the line.

Similarly, equation of a straight line which is vertical or parallel to Y-axisÂ is x = a, where a is the x-coordinate of the points on the line.

For example, the equation of the line which is parallel to X-axisÂ and contains the point (2,3) is y= 3.

Similarly, the equation of the line which is parallel to Y-axisÂ and contains the point (3,4)Â is xÂ = 3.

2. Point-slope form equation of line

Consider a non-vertical line LÂ whose slope is m, A(x,y)Â be an arbitrary point on the line and

$$\begin{array}{l}P(x_1,y_1)\end{array}$$
be the fixed point on the same line.

Slope of the line by the definition is,

$$\begin{array}{l}m\end{array}$$
=
$$\begin{array}{l}\frac{y~-~y_1}{x~-~x_1}\end{array}$$

$$\begin{array}{l}y~-~y_1\end{array}$$
=
$$\begin{array}{l}m(x~-~x_1)\end{array}$$

For example, equation of the straight line having a slope

$$\begin{array}{l}m\end{array}$$
=
$$\begin{array}{l}2\end{array}$$
and passes through the point
$$\begin{array}{l}(2,3)\end{array}$$
is

y – 3Â = 2(x – 2)

y= 2x-4+3

2x-y-1Â = 0

3. Two-point form equation of line

Let P(x,y)Â be the general point on the line LÂ which passes through the points

$$\begin{array}{l}A(x_1,y_1)\end{array}$$
and
$$\begin{array}{l}B(x_2,y_2)\end{array}$$
.

Since the three points are collinear,

slope of PAÂ = slope of AB

$$\begin{array}{l}\frac{y~-~y_1}{x~-~x_1}\end{array}$$
=
$$\begin{array}{l}\frac{y_2~-~y_1}{x_2~-~x_1}\end{array}$$

$$\begin{array}{l}y – y_{1} = (y_{2} – y_{1}). \frac{x – x_{1} }{x_{2} – x_{1}}\end{array}$$

4. Slope-intercept form equation of line

Consider a line whose slope is

$$\begin{array}{l}m\end{array}$$
which cuts the
$$\begin{array}{l}Y\end{array}$$
-axis at a distance ‘a’ from the origin. Then the distance a is called the
$$\begin{array}{l}y\end{array}$$
– intercept of the line. The point at which the line cuts
$$\begin{array}{l}y\end{array}$$
-axis will be
$$\begin{array}{l}(0,a)\end{array}$$
.

Then, equation of the line will be

y-aÂ = m(x-0)

yÂ = mx+a

Similarly, aÂ straight line having slope m cuts the X-axis at a distance b from the origin will be at the point (b,0). The distance b is called x- intercept of the line.

Equation of the line will be:

y = m(x-b)

5. Intercept form

Consider a line LÂ having x– intercept a and y– intercept b, then the line touches X– axis at (a,0)Â and Y– axis at (0,b).

By two-point form equation,

$$\begin{array}{l}y~-~0\end{array}$$
=
$$\begin{array}{l}\frac{b-0}{0~-~a} (x~-~a)\end{array}$$

$$\begin{array}{l}y\end{array}$$
=
$$\begin{array}{l}-\frac{b}{a} (x~-~a)\end{array}$$

$$\begin{array}{l}y\end{array}$$
=
$$\begin{array}{l}\frac{b}{a} (a~-~x)\end{array}$$

$$\begin{array}{l}\frac{x}{a} ~+ ~\frac{y}{b} \end{array}$$
=
$$\begin{array}{l}1\end{array}$$

For example, equation of the line which has

$$\begin{array}{l}x\end{array}$$
– intercept
$$\begin{array}{l}3\end{array}$$
and
$$\begin{array}{l}y\end{array}$$
– intercept
$$\begin{array}{l}4\end{array}$$
is,

$$\begin{array}{l}\frac{x}{3} ~+ ~\frac{y}{4}\end{array}$$
=
$$\begin{array}{l}1\end{array}$$

$$\begin{array}{l}4x~ + ~3y\end{array}$$
=
$$\begin{array}{l}12\end{array}$$

6. Normal form

Consider a perpendicular from the origin having length lÂ to line L and it makes an angle Î²Â with the positive X-axis.

Let OP be the perpendicular from the origin to the line L.
Then,

$$\begin{array}{l}OQ\end{array}$$
=
$$\begin{array}{l}l~ cosÎ²\end{array}$$

$$\begin{array}{l}PQ\end{array}$$
=
$$\begin{array}{l}l~ sinÎ²\end{array}$$

Coordinates of the point

$$\begin{array}{l}P\end{array}$$
are;
$$\begin{array}{l}P(l ~cos~Î²,l ~sin~Î²)\end{array}$$

slope of the line

$$\begin{array}{l}OP\end{array}$$
is
$$\begin{array}{l}tan~Î²\end{array}$$

Therefore,

$$\begin{array}{l}Slope~ of ~the ~line ~L\end{array}$$
=
$$\begin{array}{l}-\frac{1}{tan~Î²}\end{array}$$
=
$$\begin{array}{l}-\frac{cos~Î²}{sin~Î²}\end{array}$$

Equation of the line

$$\begin{array}{l}L\end{array}$$
having slope
$$\begin{array}{l}-\frac{cos~Î²}{sin~Î²}\end{array}$$
and passing through the point
$$\begin{array}{l}(l ~cos~Î²,l ~sin~Î²)\end{array}$$
is,

$$\begin{array}{l}y~-~l~ sin~Î²\end{array}$$
=
$$\begin{array}{l}-\frac{cos~Î²}{sin~Î²} (x~-~l ~cosÎ²)\end{array}$$

$$\begin{array}{l}y ~sin~Î²~-~l~ sin^2~ Î²\end{array}$$
=
$$\begin{array}{l}-x ~cos~Î²~+~l~cos^2~Î²\end{array}$$

$$\begin{array}{l}x~ cos~Î²~ + ~y ~sin~Î²\end{array}$$
=
$$\begin{array}{l}l(sin^2~Î² ~+ ~cos^2~Î²)\end{array}$$

$$\begin{array}{l}x ~cos~Î²~ + ~y ~sin~Î²\end{array}$$
=
$$\begin{array}{l}l\end{array}$$

You have learnt about the different forms of equation of a straight line. To know more about straight lines and its properties, log onto www.byjus.com.’