## Point-Slope Equation of a Line

### Equations of straight line in different forms

**(1) Slope form:**

Equation of a line through the origin and having slope m is **y = mx**.

**(2) One point form or Point slope form:**

Equation of a line through the point (x_{1}, y_{1}) and having slope m is **y − y _{1 }= m(x − x_{1})**.

**(3) Slope intercept form:**

Equation of a line (non-vertical) with slope m and cutting off an intercept c on the y-axis is

**y = mx + c**.

The equation of a line with slope m and the x-intercept d is y = m(x − d).

**(4) Intercept form:**

If a straight line cuts x-axis at A and the y-axis at B then OA and OB are known as the intercepts of the line on x-axis and y-axis respectively.

Then, equation of a straight line cutting off intercepts a and b on x–axis and y–axis respectively is \(\frac { x }{ a } +\frac { y }{ b } =1\).

If given line is parallel to X axis, then X-intercept is undefined.

If given line is parallel to Y axis, then Y-intercept is undefined.

**(5) Two point form:**

Equation of the line through the points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is, \((y-{ y }_{ 1 })=\frac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } (x-{ x }_{ 1 })\).

In the determinant form it is gives as \(\left| \begin{matrix} x & y & 1 \\ { x }_{ 1 } & { y }_{ 1 } & 1 \\ { x }_{ 2 } & { y }_{ 2 } & 1 \end{matrix} \right| =0\) is the equation of line.

**(6) Normal or perpendicular 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.

**(7) Symmetrical or parametric or distance form of the line:**

Equation of a line passing through (x_{1}, y_{1}) and making an angle θ with the positive direction of x-axis is ,

where r is the distance between the point P(x, y) and A(x_{1}, y_{1}).

The co-ordinates of any point on this line may be taken as (x_{1} ± r cos θ, y_{1} ± r sin θ), known as parametric co-ordinates. ‘r’ is called the parameter.