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अध्याय 9: सरल रेखाएं (Straight Lines)

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CBSE • कक्षा XI • Mathematics • अध्याय 9

अनुमानित समय: 45 Mins

प्रगति: अध्ययनरत

सरल रेखाएं (Straight Lines)

In Class 11 Mathematics, "Straight Lines" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें

प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

📐 Have You Ever Wondered?

How do computer graphics engines render laser beams or calculate the shortest distance from an airplane to a landing strip? Linear coordinate geometry...

How do computer graphics engines render laser beams or calculate the shortest distance from an airplane to a landing strip? Linear coordinate geometry unites Euclidean geometry with algebraic equations.

यह अध्याय क्यों महत्वपूर्ण है

In Class 11 Mathematics, "Straight Lines" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्ययन से पूर्व (आवश्यक ज्ञान)

Cartesian coordinates and distance formula from Class 10.
Section formula.
Slope of a line.

इस अध्याय के लक्ष्य

Define Slope (Gradient) of a non-vertical line: $m = \tan\theta = \frac{y_2 - y_1}{x_2 - x_1}$.
State conditions for parallelism ($m_1 = m_2$) and perpendicularity ($m_1 m_2 = -1$).
Write equations of lines in various forms: Point-Slope, Two-Point, Slope-Intercept ($y = mx + c$), and Intercept form ($\frac{x}{a} + \frac{y}{b} = 1$).
Find the angle between two intersecting lines: $\tan\theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right|$.
Calculate the perpendicular distance from a point to a line: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.

अध्याय रूपरेखा एवं प्रगति

1 1. Slope & Angle Between Lines

2 2. Standard Forms of Line Equations

3 3. Distance of a Point from a Line

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Slope & Angle Between Lines

The Slope $m$ of a line making angle $\theta$ with the positive x-axis is: $$\mathbf{m = \tan\theta = \frac{y_2 - y_1}{x_2 - x_1}} \quad (x_1 \ne x_2)$$
• Parallel lines: $\mathbf{m_1 = m_2}$.
• Perpendicular lines: $\mathbf{m_1 \cdot m_2 = -1}$.
• Angle $\theta$ between two lines: $\mathbf{\tan\theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right|}$.

2. Standard Forms of Line Equations

Slope-Intercept Form: $\mathbf{y = mx + c}$ ($c$ is y-intercept).
Point-Slope Form: $\mathbf{y - y_1 = m(x - x_1)}$.
Two-Point Form: $\mathbf{y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)}$.
Intercept Form: $\mathbf{\frac{x}{a} + \frac{y}{b} = 1}$ ($a, b$ are intercepts on coordinate axes).

3. Distance of a Point from a Line

The perpendicular distance $d$ from point $(x_1, y_1)$ to line $Ax + By + C = 0$ is: $$\mathbf{d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}}$$ Distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$: $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.

चित्रात्मक व्याख्या एवं मॉडल

Straight Lines Master Matrix

Conceptual framework, core mechanisms, and analytical relationships

Academic Architecture

1. Slope & Angle Between Lines • 2. Standard Forms of Line Equations

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

Slope / Gradient: Tangent of inclination measuring rate of elevation.

मुख्य बिंदु 2

Perpendicular Orthogonality: Slopes are negative reciprocals ($m_1 m_2 = -1$).

मुख्य बिंदु 3

Intercept Equation: Symmetric coordinate representation $x/a + y/b = 1$.

मुख्य बिंदु 4

Perpendicular Distance: Exact normal projection distance formula from coordinate to line.

मुख्य बिंदु 5

Parallel Separation: Distance between parallel lines determined strictly by constant difference.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

Find the slope of a line passing through the points $(3, -2)$ and $(-1, 4)$.

उत्तर एवं व्याख्या देखें

उत्तर: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{-1 - 3} = \frac{6}{-4} = -\frac{3}{2}$.
m = -3/2.

Find the equation of a line passing through $(-4, 3)$ with slope $1/2$.

उत्तर एवं व्याख्या देखें

उत्तर: Point-slope form: $y - 3 = \frac{1}{2}(x - (-4)) \implies 2y - 6 = x + 4 \implies x - 2y + 10 = 0$.
x - 2y + 10 = 0.

Find the angle between the lines $y - \sqrt{3}x - 5 = 0$ and $\sqrt{3}y - x + 6 = 0$.

उत्तर एवं व्याख्या देखें

उत्तर: $m_1 = \sqrt{3}$, $m_2 = \frac{1}{\sqrt{3}}$. $\tan\theta = |\frac{1/\sqrt{3} - \sqrt{3}}{1 + 1}| = |\frac{-2/\sqrt{3}}{2}| = \frac{1}{\sqrt{3}} \implies \theta = 30^\circ$ or $150^\circ$.
30° (or 150°).

Find the distance of the point $(3, -5)$ from the line $3x - 4y - 26 = 0$.

उत्तर एवं व्याख्या देखें

उत्तर: $d = \frac{|3(3) - 4(-5) - 26|}{\sqrt{3^2 + (-4)^2}} = \frac{|9 + 20 - 26|}{\sqrt{25}} = \frac{|3|}{5} = \frac{3}{5}\text{ units}$.
3/5 units.

Find the distance between the parallel lines $3x - 4y + 9 = 0$ and $3x - 4y + 7 = 0$.

उत्तर एवं व्याख्या देखें

उत्तर: $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} = \frac{|9 - 7|}{\sqrt{3^2 + (-4)^2}} = \frac{2}{5}\text{ units}$.
2/5 units.

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