Straight lines and Family of Straight lines Summary

• The equation of a straight line parallel to x-axis and at a distance h from it is given by y = h.

• The equation of the straight line parallel to y-axis and at a distance k from it is given by x = k.

• The equation of the straight line having slope m and intercept on

y-axis as c is given by y = mx+c. (Slope-Intercept form)

• The equation of the straight line having intercepts a and b on x-axis and y-axis respectively is given by (Intercept form)

• The equation of the straight line passing through the points (x₁, y₁) is given by

           (Two-point form)

• Here, we assume that x₁ ¹ x₂; in case x₁ = x₂, then the line is vertical and its equation is x = x₁ (or x₂)

• The equation of the straight line passing through (x1, y1) and making angle q with the positive direction of x-axis is given by (Distance form)

         Where r is the distance between the points (x, y) and (x₁, y₁).

• The equation of a straight line for which the perpendicular from the origin makes an angle a and is of length p, is given by x cos a + y sin a = p. (Normal form)

• Every straight line has an equation of the form ax+by+c=0 and conversely an equation of the type ax+by+c=0 (a,b are both not equal to zero) always represents a straight line.

• Two lines are said to be intersecting if there is exactly one point which is common to both lines.

• The tangent of the acute angle between two straight lines with slopes m₁ and m₂ is given by

• Three lines are said to be concurrent if all the three lines passes through a point. The common point of the concurrent lines is called the point of concurrence.

• The orthocentre of a triangle is the point of concurrence of the altitudes drawn from the vertices to the opposite sides of the triangle.

• The circumcentre of a triangle is the point of concurrence of the right bisectors of the sides of the triangle.

• The length of perpendicular of the point (x₁,y₁) from the straight line ax + by + c = 0 is equal to

• A set of lines satisfying a given condition is called a family of lines. A family of lines can be represented by a linear equation in x and y and involving one arbitrary constant, which is called the parameter of the family of lines under consideration.