The slope of the line through (1, 2) and (3, 8) is:

3. m = (8 − 2)/(3 − 1) = 3.

The slope of a line perpendicular to a line of slope 2 is:

−1/2. m₁m₂ = −1 gives m₂ = −1/2.

The equation of the line with slope 3 and Y-intercept −2 is:

y = 3x − 2. Slope-intercept form y = mx + c.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 1 Chapter 5: Locus and Straight Line

Short notes, key concepts, quick revision and practice MCQs with answers

A locus is a set of points satisfying a geometric condition. The chapter finds equations of loci, defines the inclination and slope of a line, gives the conditions for parallel and perpendicular lines and the angle between two lines, and writes the equation of a line in different forms, ending with distances of a point from a line and between parallel lines.

Key concepts: Locus and Straight Line

Locus and its equation: A locus is a set of points in a plane satisfying given conditions. Its equation is satisfied by the coordinates of every point on the locus and by no other point.
Inclination and slope: The inclination θ is the angle a line makes with the positive X-axis (0° ≤ θ < 180°). Slope m = tan θ = (y₂ − y₁)/(x₂ − x₁). A vertical line has no defined slope; a horizontal line has slope 0.
Parallel and perpendicular lines: Non-vertical lines are parallel if m₁ = m₂ and perpendicular if m₁ × m₂ = −1.
Angle between two lines: The acute angle θ between lines with slopes m₁ and m₂ is given by tan θ = |(m₁ − m₂)/(1 + m₁m₂)|, provided 1 + m₁m₂ ≠ 0.
Forms of the equation of a line: Slope-point: y − y₁ = m(x − x₁). Slope-intercept: y = mx + c. Two-point: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁). Double intercept: x/a + y/b = 1. Normal: x cos α + y sin α = p.
General form: ax + by + c = 0. Slope = −a/b, X-intercept = −c/a, Y-intercept = −c/b.
Distances: Distance of (x₁, y₁) from ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²). Distance between parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0 is |c₁ − c₂| / √(a² + b²).

Quick revision: important points

Equation of the X-axis is y = 0 and of the Y-axis is x = 0.
A line through the origin has equation y = mx.
Lines parallel to ax + by + c = 0 have the form ax + by + k = 0; perpendicular lines have the form bx − ay + k = 0.
Distance of the origin from ax + by + c = 0 is |c| / √(a² + b²).
Three points are collinear if the slope between any two pairs is the same.
The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.

Practice MCQs with answers

The slope of the line through (1, 2) and (3, 8) is:
1. 2
2. 3
3. 1/3
4. 6
Show answer
Answer: B. 3
m = (8 − 2)/(3 − 1) = 3.
The slope of the line 2x + 3y − 6 = 0 is:
1. 2/3
2. −2/3
3. 3/2
4. −3/2
Show answer
Answer: B. −2/3
m = −a/b = −2/3.
The slope of a line perpendicular to a line of slope 2 is:
1. 2
2. −2
3. 1/2
4. −1/2
Show answer
Answer: D. −1/2
m₁m₂ = −1 gives m₂ = −1/2.
The equation of the line with slope 3 and Y-intercept −2 is:
1. y = 3x + 2
2. y = −2x + 3
3. y = 3x − 2
4. 3y = x − 2
Show answer
Answer: C. y = 3x − 2
Slope-intercept form y = mx + c.
The X-intercept of 3x + 4y = 12 is:
1. 3
2. 4
3. 12
4. −4
Show answer
Answer: B. 4
Put y = 0: 3x = 12, x = 4.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters