Mathematics > Straight lines

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  • Distance between the points  is

  • The coordinates of a point dividing the line segment joining the points    internally, in the ratio m: n are

  • If , the coordinates will be.
  • Area of the triangle whose vertices are  is
  • If the area of the triangle ABC is zero, then three points A, B and C lie on a line, i.e., they are collinear.
  • If  is the inclination of a line l, then  is called the slope or gradient of the line l. The slope of a line whose inclination is 90° is not defined. The slope of a line is denoted by m. Thus,
  • Slope
  • If the line  is parallel to  , then their inclinations are equal.
  • If the lines  and  are perpendicular such that  makes an angle and  makes an angle  with the x-axis,  then
  • Two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other
  • An acute angle (say θ) between lines L1 and L2 with slopes  is given by

  • Two lines are parallel if and only if their slopes are equal.
  • Two lines are perpendicular if and only if product of their slopes is –1.
  • Three points A, B and C are collinear, if and only if slope of AB = slope of BC.
  • Equation of the horizontal line having distance a from the x-axis is either

  • Equation of the vertical line having distance b from the y-axis is either

  • The point (x, y) lies on the line with slope m and through the fixed point  if and only if its coordinates satisfy the equation
  • Equation of the line passing through the points  is given by

  • The point (x, y) on the line with slope m and y-intercept c lies on the line if and only if
  • If a line with slope m makes x-intercept d. Then equation of the line is
  • Equation of a line making intercepts a and b on the x-and y-axis, respectively, is

  • The equation of the line having normal distance from origin p and angle between normal and the positive x-axis  is given by  .
  • Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the general linear equation or general equation of a line.
  • The perpendicular distance (d) of a line Ax + By+ C = 0 from a point  is given by

  • Distance between the parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0, is given by

 

Examples

Question

If the angle between two lines is π/4 and slope of one of the lines is 1/2, find the slope of the other line.

Solution

θ = π/4

Substituting values,

 

Question

Three points P (h, k), Q (x1, y1) and R (x2, y2) lie on a line. Show that (h – x1) (y2 – y1) = (k – y1) (x2 – x1).

Solution

Since points P, Q and R are collinear, we have Slope of PQ = Slope of QR

 

Question

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

Solution

Here m = – 4 and given point  is (– 2, 3). By slope-intercept form formula, equation of the given line is

y – 3 = – 4 (x + 2) or 4x + y + 5 = 0, which is the required equation.

 

Question

Equation of a line is 3x – 4y + 10 = 0. Find its (i) slope, (ii) x – and y-intercepts.

Solution

   

  1. Given equation 3x – 4y + 10 = 0 can be written as

    Comparing with y = mx + c, we have slope of the given line as m = 4/3

  2. Given equation can be written as

    y intercept is 5/2.

 

Question

Find the equation of a line perpendicular to the line x − 2y + 3 = 0 and passing through the point (1, – 2).

Solution

Given line x − 2 y + 3 = 0 can be written as

y = x/2 + 3/2

Slope of the line (1) is m1 = 2. Therefore, slope of the line perpendicular to line (1) is m2 = -1/m1 = -1/2

Equation of the line with slope – 2 and passing through the point (1, – 2) is y − (− 2) = −2(x −1) or y= − 2x.

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