# 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 (x1, y1) and (x2, y2) 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 (x1, y1) is y – y1 = m(x – x1)
• Two point form: y – y1 = $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ (x – x1)
• 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 m1 = m2
• Two lines are perpendicular if, m1m2 = -1

4. Coincident lines

Two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 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 (x1, y1) and making an angle α with y = mx + c

y – y1 = $$\frac{m \mp \tan \alpha}{1 \pm m \tan \alpha}$$ (x – x1)

6. Length of perpendicular

rom (x1, y1) 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 + c1 = 0 and ax + by + c2 = 0 then d = $$\frac{\left|c_{1}-c_{2}\right|}{\sqrt{a^{2}+b^{2}}}$$

8. Condition of concurrency

For the lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, a3x + b3y + c3 = 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

a1x + b1y + c1 = 0 and a2x + b2y + c2 = 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 c1 > 0, c2 > 0

 Condition Acuteangle bisector obtuse angle bisector a1a2 + b1b2 > 0 – + a1a2 + b1b2 < 0 + –

10. Homogeneous equation

• If y = m1x and y = m2x be the two equation represented by
ax2 + 2hxy + by2 = 0 then
m1 + m2 = –$$\frac{2h}{b}$$, m1m2 = $$\frac{a}{b}$$
• 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, ax2 + 2hxy + by2 = 0 is $$\frac{x^{2}-y^{2}}{a-b}=\frac{x y}{h}$$

11. General equation of second degree

ax2 + 2hxy + by2 + 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,
h2 > ab → Hyperbola
h2 = ab → Parabola
h2 < 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 a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 can be represented by the equation is
(a1 + b1y + c1) + λ (a2x + b2y + c2) = 0