# Straight Line Formulas

The Concept of Straight Lines is Crucial in Coordinate Geometry. You might feel it difficult to understand the concept and solve numerous problems involving Straight Line. To help you out we have compiled Straight Lines Formulae to solve different problems on Straight Lines in Coordinate Geometry. You can check out both basic and advanced formulas of Straight Lines and get a good grip on the concept.

## Straight Lines Formulae List | Cheat Sheet & Tables for Straight Lines Formulas

The below listed Straight Formulae can save your time and you can solve the problems simply. Use them during your calculations and find the solution for your Straight Line Problems. Memorize the Straight Line Formulae List existing and overcome the tedious task of performing lengthy calculations. You will be familiar with the concept of Straight Lines in a better way after referring to the Straight Lines Formula Cheat Sheet & Tables provided.

**1. Slope of a line**

- m = tan θ, where θ is the angle made by a line with the positive direction of x axis in anticlockwise
- The slope of a line joining two points (x
_{1}, y_{1}) and (x_{2}, y_{2}) is given by m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

**2. Different forms of the equation of straight line**

- Slope – Intercept form: y = mx + c.
- Slope point form: The equation of a line with slope m and passing through a point (x
_{1}, y_{1}) is y – y_{1}= m(x – x_{1}) - Two point form: y – y
_{1}= \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) (x – x_{1}) - Intercept Form: \(\frac{x}{a}+\frac{y}{b}=1\)
- Normal (perpendicular) form of a line: x cosα + y sinα = p
- Parametric form (distance form): \(\frac{x-x_{1}}{\cos \theta}=\frac{y-y_{1}}{\sin \theta}=r\)

**3. The angle between two straight lines**

- tan θ = \(\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|\)
- Two lines are parallel if m
_{1}= m_{2} - Two lines are perpendicular if, m
_{1}m_{2}= -1

**4. Coincident lines**

Two lines a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 are coincident only if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

**5. Equation of straight lines through (x _{1}, y_{1}) and making an angle α with y = mx + c**

y – y_{1} = \(\frac{m \mp \tan \alpha}{1 \pm m \tan \alpha}\) (x – x_{1})

**6. Length of perpendicular**

rom (x_{1}, y_{1}) to the straight line ax + by + c = 0 then

p = \(\frac{\left|a x_{1}+b y_{1}+c\right|}{\sqrt{a^{2}+b^{2}}}\)

**7. Distance between two parallel lines**

ax + by + c_{1} = 0 and ax + by + c_{2} = 0 then d = \(\frac{\left|c_{1}-c_{2}\right|}{\sqrt{a^{2}+b^{2}}}\)

**8. Condition of concurrency**

For the lines a_{1}x + b_{1}y + c_{1} = 0, a_{2}x + b_{2}y + c_{2} = 0, a_{3}x + b_{3}y + c_{3} = 0

\(\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{array}\right|=0\)

**9. Bisectors of angles between two lines**

a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0

\(\frac{a_{1} x+b_{1} y+c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}=\pm \frac{a_{2} x+b_{2} y+c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\)

Condition to find acute angle bisector, obtuse angle bisector, when c_{1} > 0, c_{2} > 0

Condition | Acuteangle bisector | obtuse angle bisector |

a_{1}a_{2} + b_{1}b_{2} > 0 |
– | + |

a_{1}a_{2} + b_{1}b_{2} < 0 |
+ | – |

**10. Homogeneous equation**

- If y = m
_{1}x and y = m_{2}x be the two equation represented by

ax^{2}+ 2hxy + by^{2}= 0 then

m_{1}+ m_{2}= –\(\frac{2h}{b}\), m_{1}m_{2}= \(\frac{a}{b}\) - If θ is the acute angle between the pair of straight lines then tanθ = \(\left|\frac{2 \sqrt{h^{2}-a b}}{a+b}\right|\)
- The equation of the straight lines bisecting the angles between the straight lines, ax
^{2}+ 2hxy + by^{2}= 0 is \(\frac{x^{2}-y^{2}}{a-b}=\frac{x y}{h}\)

**11. General equation of second degree**

ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0

(a) represent a pair of two straight lines if

∆ = \(\left|\begin{array}{lll}a & h & g \\h & b & f \\g & f & c\end{array}\right|\) = 0.

(b) represent a circle if ∆ ≠ 0, a = b, h = 0

(c) Represent conic section if ∆ ≠ 0, a ≠ b,

h^{2} > ab → Hyperbola

h^{2} = ab → Parabola

h^{2} < ab → Ellipse

**12. “Q” is foot of perpendicular**

\(\frac{\alpha-x_{1}}{a}=\frac{\beta-y_{1}}{b}=-\frac{\left(a x_{1}+b y_{1}+c\right)}{a^{2}+b^{2}}\)

“B” is image of “A”

\(\frac{x_{2}-x_{1}}{a}=\frac{y_{2}-y_{1}}{b}=-\frac{2\left(a x_{1}+b y_{1}+c\right)}{a^{2}+b^{2}}\)

**13. Family of lines**

Any line passing through the point of intersection of the lines a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 can be represented by the equation is

(a_{1} + b_{1}y + c_{1}) + λ (a_{2}x + b_{2}y + c_{2}) = 0