9. Straight Lines

9 topics2 h

Chapter 9 of NCERT Mathematics for Class 11: Straight Lines. Three sections — Slope, Writing the equation of a line and How far away a line is. Nine videos, 2 h in all.

Worked answers to the 70 exercise questions in this chapter: Exercise 9.1 · Exercise 9.2 · Exercise 9.3 · Miscellaneous Exercise

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Part 1

Slope

14 minSteepness as the tangent of an angle, and the one line that has noneTip a line toward vertical from one side and its slope races past every number that can be named, then off the opposite way from the other side. It has no slope - not an infinite one.14 min · 14 parts15 minEqual slopes mean parallel; slopes multiplying to minus one mean perpendicularEqual slopes make two lines parallel, and slopes multiplying to minus one make them perpendicular - except the axes, the one perpendicular pair whose product can never even be formed.15 min · 15 parts15 minRecovering the angle between two lines from their two slopesTwo crossing lines make four angles, not one, in only two distinct sizes. 'The angle between two lines' is not a complete question until something decides which size is meant.15 min · 17 parts

Part 2

Writing the equation of a line

12 minLines parallel to an axis, where one coordinate never changesy = 3 carries no x at all - not a missing ingredient, but a statement that the first coordinate is free. A vertical line has no slope, so no slope-based method can ever write it.12 min · 14 parts15 minOne point and a slope, or two points: the same condition written twicePoint-slope and two-point look like two formulas to remember. They are one condition in two coats, and clearing the fraction repairs the one point the quotient could never discuss.15 min · 17 parts15 minNaming a line by where it crosses the axesA line's own crossings of the axes hand it two free points - except lines through the origin or parallel to an axis, which have no intercepts, and no equation of this shape.15 min · 16 parts14 minNaming Ax + By + C = 0, the one form the distance formula ahead will takeEvery slope-based way of writing a line refuses a vertical one. A sixth form, Ax + By + C = 0, is the roof over all five, refusing only to describe the whole plane, or nothing.14 min · 13 parts

Part 3

How far away a line is

16 minThe shortest route from a point to a line, read off its coefficientsThe formula for a point's distance from a line never finds where the perpendicular lands - it builds a triangle from the point and the line's two axis crossings, and measures the area two different ways.16 min · 15 parts12 minThe gap between two parallel lines as one distance measured onceStand anywhere on one of two parallel lines and measure across to the other, and the answer never changes - though the usual proof only ever measures from one convenient point.12 min · 13 parts

What you will be able to do

  • State what the inclination of a line is, and give the closed range it lies in
  • State the parallelism criterion of §9.2.2 in your own words, as two claims rather than one
  • Explain why a crossing of two lines offers two candidate angles and why they are supplementary
  • State what §9.3 asks an equation of a line to do, in both directions
  • Derive the point-slope equation from the slope of a segment joining a fixed point to a variable one
  • Derive the slope-intercept equation by applying the point-slope form at the line's y-axis crossing
  • State what the general linear equation is and what its coefficients are allowed to be
  • State what the chapter takes a point's distance from a line to mean
  • Say what has to be true before "the distance between two parallel lines" is a well-defined quantity

What this chapter assumes you already know

  • Plotting points in the plane, and the four coordinate results §9.1 recalls: distance between two points, the section formula, the mid-point as its equal-ratio case, and the coordinate expression for the area of a triangle
  • The tangent function on the interval from zero to a straight angle, from Chapter 3 — in particular that it is positive on acute angles, negative on obtuse ones, zero at both ends, and has no value at a right angle
  • Right triangles and the tangent of an acute angle as a ratio of two sides
  • Signed differences: that a coordinate difference carries direction as well as size
  • Measuring an angle anticlockwise from a fixed ray, and the convention that fixes its sign
  • That the tangent of a right angle does not exist, and that the vertical line is therefore outside every slope statement in §9.2
  • The tangent of an angle increased by a right angle, from Chapter 3 — the identity that turns a tangent into a negative cotangent
  • Corresponding angles on a transversal cutting two parallel lines

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Slope is defined as rise over run."
  • "A vertical line has infinite slope."
  • "The inclination is whichever angle looks smaller."
  • "A negative slope means the line is below the axis."
  • "You must take the points in left-to-right order."
  • "The obtuse case needs a different formula."
  • "Zero slope and no slope are two ways of saying the same thing."
  • "Area zero just means the triangle is small."