Straight Lines Class 11 Notes - Chapter 10

The slope of a line (m) is determined by the value of tan θ, where θ is the angle made by the line with the positive direction of the x-axis in an anti-clockwise direction. The slope of the line passing through two points P (a₁, b₁) and Q (a₂, b₂) is given by:

If m1 and m2 be the slopes of two lines. The angle θ between them is given by:

In case of acute angle,

• For parallel lines, m₁ = m₂
• For perpendicular lines, m₁.m₂ = -1

Three points A (h, k), B(m₁, n₁) and C(m₂, n₂) are said to be collinear if the slope of AB = slope of BC.

i.e.

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Equation of Line – Different Forms

1. The equation of a line parallel to the x-axis and at a distance (p) from the x-axis is given by y = ± p.
2. The equation of a line parallel to the y-axis and at a distance (q) from y-axis is given by x = ± q.
3. The equation of a line [Point-slope form] having slope (m) and passing through the point (a₀, b₀) is given by y – b₀ = m(x – a₀).
4. The equation of a line [Two-point-form] passing through two points (a1, b1) and (a2, b2) is given by,

5. The equation of a line [Slope intercept form] making an intercept (p) on the y-axis (slope m) is given by y = mx + p. [value of p will be +ve or -ve based on the intercept made on
the +ve or -ve side of the y-axis].

6. The equation of the line [Intercept form] making intercepts p and q on the x and y-axis, respectively, is given by

In normal form, the equation of the line is given by x cos ω + y sin ω = p. Where p = Length of perpendicular (p) from the origin and ω = Angle, which normally makes with the +ve x-axis direction.

The points (m₁, n₁) and (m₂, n₂) are on the same or opposite side of a line px + qy + r = 0 if pm₁ + qn₁ + r and pm₂ + qn₂ + r are of the same sign or of opposite signs respectively. The lines xm₁ + yn₁ + o1 = 0 and xm₂ + yn₂ + o = 0 are perpendicular, if, m₂m₁ + n₂n₁ = 0.

Straight Lines Class 11 Practice Questions

1. Determine the equation of lines passing through (2, 3) and making an angle of 60° with
   x-axis.
2. Determine the equation of a line parallel to the line 2x + 5y = 2 and passing through the point of intersection of lines 12x + 2y = 35 and 4x + 53y + 68 = 0.
3. Determine the length of the triangle if the equation of the base of an equilateral triangle is 2x + 3y = 42 and the vertex is (3, -4).

4. Determine the value of p and q if the intercepts cutoff on the coordinate axes by the
   line px + qy + 3 = 0 are equal but have opposite signs to those cut off by the
   line x – 2y + 4 = 0 on the axes.

5. Determine the equation of the line if the intercept of a line connecting the coordinate axes is separated by a point (4, -3) in the ratio 2:4.

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