Straight Lines
Coordinate geometry me hum shapes ko algebra se padhte hain. Sabse simple shape hai straight line. Is chapter me line ka slope, line likhne ke alag-alag forms, do lines ke beech ka angle, aur point se line ki doori dekhenge.
1. Slope (Gradient) of a Line
Line ka slope \(m\) batata hai ki line kitni "tedhi" hai. \(x\)-axis se \(\theta\) angle banane wali line ka slope \(m = \tan\theta\).
Do points \((x_1, y_1)\) aur \((x_2, y_2)\) se slope:
\[ m = \frac{y_2 - y_1}{x_2 - x_1}, \qquad x_1 \neq x_2. \]
- Horizontal line: \(m = 0\).
- Vertical line: slope undefined (\(\theta = 90^\circ\)).
2. Parallel aur Perpendicular Lines
- Parallel lines ke slopes barabar: \(m_1 = m_2\).
- Perpendicular lines ke slopes ka product \(-1\): \(m_1 m_2 = -1\).
Angle between two lines
\[ \tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|. \]
3. Line ke Various Forms
| Form | Equation | Kab use karein |
|---|---|---|
| Point–slope | \(y - y_1 = m(x - x_1)\) | ek point aur slope pata ho |
| Two-point | \(y - y_1 = \dfrac{y_2 - y_1}{x_2 - x_1}(x - x_1)\) | do points pata ho |
| Slope–intercept | \(y = mx + c\) | slope aur \(y\)-intercept \(c\) |
| Intercept | \(\dfrac{x}{a} + \dfrac{y}{b} = 1\) | \(x\)-intercept \(a\), \(y\)-intercept \(b\) |
| General | \(Ax + By + C = 0\) | koi bhi line |
General form \(Ax + By + C = 0\) me slope \(= -\dfrac{A}{B}\).
Points \((1, 2)\) aur \((3, 6)\) se guzarne wali line. \[ m = \frac{6 - 2}{3 - 1} = 2, \qquad y - 2 = 2(x - 1) \Rightarrow y = 2x. \]
4. Distance of a Point from a Line
Point \((x_1, y_1)\) ki line \(Ax + By + C = 0\) se doori: \[ d = \frac{|A x_1 + B y_1 + C|}{\sqrt{A^{2} + B^{2}}}. \]
Distance between two parallel lines
\(Ax + By + C_1 = 0\) aur \(Ax + By + C_2 = 0\) ke beech:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^{2} + B^{2}}}. \]
Point \((2, 3)\) ki line \(3x + 4y - 5 = 0\) se doori: \[ d = \frac{|3(2) + 4(3) - 5|}{\sqrt{9 + 16}} = \frac{|6 + 12 - 5|}{5} = \frac{13}{5}. \]
Key Takeaways
- Slope \(m = \tan\theta = \dfrac{y_2 - y_1}{x_2 - x_1}\).
- Parallel: \(m_1 = m_2\); Perpendicular: \(m_1 m_2 = -1\).
- Forms yaad rakho: point-slope, slope-intercept \(y = mx + c\), intercept, general.
- Point se line ki doori: \(\dfrac{|Ax_1 + By_1 + C|}{\sqrt{A^{2}+B^{2}}}\).