Straight Line 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

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