Show that the product of the perpendicular drawn from the two points  upon the straight line

If p and pʹ are the perpendicular from the origin upon the straight line whose equations are and , prove that .

Find the distance between two parallel straight lines y =mx + c and y =mx + d.

What are the points on the x–axis whose perpendicular distance from the straight line  is a?

Show that the perpendiculars let fall from any point of the straight line 2x + 11y = 5 upon the two straight lines 24x + 7y =20 and 4x – 3y = 2 are equal to each other.

Find the perpendicular distance from the origin of the perpendicular from the point (1, 2) upon the straight line .

Show that the straight lines 2x − 3y + 5 = 0, 3x + 4y − 7 = 0 and 9x − 5y + 8 = 0 meet at a point.

Find the equation of the straight line which passes through the intersection of the straight lines 2x − 3y + 4 = 0, 3x + 4y − 5 = 0 and is perpendicular to the straight line 6x − 7y + 8 =0

Find the equation to the bisector of the angles between the straight lines 3x − 4y + 7 = 0 and 12x − 5y − 8 = 0.

Two straight lines cut the axis of x at a distance a and −a and axis of y at distance b and bʹ respectively. Find the coordinates of their point of intersection.

Prove that the following set of three lines meet at a point  and y = x

Find the conditions that the straight lines ,   may meet in a point.

Find the equation of the line passing through the point (a, b) and the intersection of the lines  and  

Find the equation of the line through the point of intersection of the lines x − 2y − a = 0 and x + 3y − 2a = 0 and parallel to the straight line 3x + 4y = 0

Find the equation of the line through the intersection of the lines 3x − 4y + 1 = 0 and 5x + y − 1 = 0 and cutting off equal intercepts from the axes.

Find the equation of the line through the intersection of the lines 2x − 3y = 10 and x + 2y = 6 and the intersection of the lines 16x − 10 y = 33 and 12x + 14y + 29 = 0

Show that the diagonals of the parallelogram formed by the four straight lines  are perpendicular to each other.

One side of a square is inclined to the x–axis at an angle α and one of its extremities is at origin. Prove that the equation of diagonals are
y(cosα − sinα) = x(sinα + cosα)
y(cosα + sinα) + x(cosα − sinα) = a

Find the equations to the straight lines bisecting the angles between the pair of straight lines, placing first the bisector of the angles in which the origin lies,  

If the equation  represents two straight lines, prove that the square of the distance of the point of intersection from the origin is

Prove that the product of perpendiculars let fall from the point  upon the straight lines  is

What does the equation  become when it is transformed to parallel axes through the point

What does the equation  become if the origin is moved to the point