What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

Define the slope of a line.

If a line makes inclination \(\theta\) with the positive \(x\)-axis, its slope is \(m=\tan\theta\) (undefined for a vertical line).

Give the slope formula through two points.

For \((x_1,y_1)\) and \((x_2,y_2)\) with \(x_1\ne x_2\), \(m=\dfrac{y_2-y_1}{x_2-x_1}\).

State the conditions for two lines to be parallel and perpendicular.

Parallel iff \(m_1=m_2\); perpendicular iff \(m_1 m_2=-1\) (slopes are negative reciprocals).

Write the point-slope form of a line.

A line of slope \(m\) through \((x_0,y_0)\) is \(y-y_0=m(x-x_0)\).

Find the equation of the line through \((-2,3)\) with slope \(-4\).

\(y-3=-4(x+2)\Rightarrow 4x+y+5=0\).

Write the intercept form of a line.

A line with \(x\)-intercept \(a\) and \(y\)-intercept \(b\) is \(\dfrac{x}{a}+\dfrac{y}{b}=1\).

Give the formula for the angle between two lines.

The acute angle \(\theta\) satisfies \(\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1 m_2}\right|\), provided \(1+m_1 m_2\ne0\).

State the distance of a point from a line.

The distance of \((x_1,y_1)\) from \(Ax+By+C=0\) is \(d=\dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}\).

Find the distance between \(3x-4y+7=0\) and \(3x-4y+5=0\).

Parallel lines: \(d=\dfrac{|7-5|}{\sqrt{3^2+4^2}}=\dfrac{2}{5}\).

How is collinearity of three points checked using slopes?

Points \(A,B,C\) are collinear iff slope of \(AB=\) slope of \(BC\) (equivalently the area of triangle \(ABC\) is zero).

Straight Lines

Class 11 Mathematics · Chapter 9

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CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

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CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

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Summary

Coordinate geometry represents the straight line algebraically, building on the distance, section and area formulae from earlier classes. The inclination \(\theta\) of a line gives its slope \(m=\tan\theta\), and for two points the slope is \(m=\dfrac{y_2-y_1}{x_2-x_1}\). Two lines are parallel iff their slopes are equal and perpendicular iff \(m_1 m_2=-1\); the acute angle between them satisfies \(\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1 m_2}\right|\). The chapter derives several forms of a line's equation: point-slope \(y-y_0=m(x-x_0)\), two-point, slope-intercept \(y=mx+c\), intercept \(\dfrac{x}{a}+\dfrac{y}{b}=1\), and the general form \(Ax+By+C=0\). The distance of a point \((x_1,y_1)\) from the line \(Ax+By+C=0\) is \(\dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}\), and the distance between parallel lines \(Ax+By+C_1=0\) and \(Ax+By+C_2=0\) is \(\dfrac{|C_1-C_2|}{\sqrt{A^2+B^2}}\). Special cases include horizontal lines \(y=a\) and vertical lines \(x=b\), while three points are collinear iff the slope of \(AB\) equals the slope of \(BC\) (equivalently, the area of triangle \(ABC\) is zero). Any first-degree equation \(Ax+By+C=0\), with \(A,B\) not both zero, represents a straight line and can be reduced to any of these standard forms. These tools handle collinearity, concurrency of lines, reflections of a point in a line, and many geometric and real-life applications such as relating two linearly varying quantities.

Slope of a line and inclinationConditions for parallel and perpendicular linesAngle between two linesVarious forms of the equation of a lineGeneral equation of a lineDistance of a point from a line and between parallel lines

Key terms

Slope (gradient)

\(m=\tan\theta\) where \(\theta\) is the inclination; for two points \(m=\dfrac{y_2-y_1}{x_2-x_1}\).

Parallel and perpendicular

Parallel iff \(m_1=m_2\); perpendicular iff \(m_1 m_2=-1\).

Point-slope form

\(y-y_0=m(x-x_0)\).

Slope-intercept form

\(y=mx+c\), where \(c\) is the \(y\)-intercept.

Intercept form

\(\dfrac{x}{a}+\dfrac{y}{b}=1\), with \(x\)-intercept \(a\) and \(y\)-intercept \(b\).

Distance of point from line

\(d=\dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}\) for line \(Ax+By+C=0\).

Extra questions & answers

Try one from this chapter

The slope of a line with inclination \(\theta\) is:

Flashcards & interactive practice