Class 11th mathematics(straight lines)

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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 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:
m=tanθ=b2−b1a2−a1
If m1 and m2 be the slopes of two lines. The angle θ between them is given by:
tanθ=±(m1−m2)1+m1m2
In case of acute angle, tanθ=±(m1−m2)1+m1m2
  • 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. n1−km1−h=n2−n1m2−m1Or, (k−n1)(m2−m1)=(h−m1)(n2−n1)

Equation of Line – Different Forms

  1. The equation of a line parallel to the x-axis and at a distance (p) from 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 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,
y−b1=(b2−b1a2−a1)(x−a1) 
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].
  1. The equation of the line [Intercept form] making intercepts p and q on x and y axis respectively is given by
xa+yb=1
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 normal 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.

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