The family of lines x(a + 2b) + y(a + 3b) = a + b passes through the point for all values of a and b. Find the point.
The given equation can be written as
a(x + y – 1) + b(2x + 3y – 1) = 0
which is equation of a line passing through the point of intersection of the lines x + y – 1 = 0 and 2x + 3y – 1 = 0. The point of intersection of these lines is (2, –1). Hence the given family of lines passing through the point (2, –1) for all values of a and b.
A straight canal is at a distance of km from a city and the nearest path from the city to the canal is in the north-east direction. Find whether a village which is at 3 km north and 4 km east from the city lies on the canal or not. If not, then on which side of the canal is the village situated?
Find the general equation of the line which is parallel to
3x – 4y + 5 = 0. Also find such line through the point (–1, 2).
Find the general equation of the line which perpendicular to x + y + 4 = 0. Also find such line through the point (1, 2).
Find the sum of the abscissas of all the points on the line x + y = 4 that lie at a unit distance from the line 4x + 3y – 10 = 0.
If p and p’ are the length of the perpendiculars from the origin to the straight line whose equations are , then find the value of 4p2 + p’2.
Find the distance between the lines 5x – 12y + 2 = 0 and
5x – 12y – 3 = 0.
Find the equations of the line parallel to 5x – 12y + 26 = 0 and at a distance of 4 units from it.
If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 (a, b and cbeing distinct and difference from 1) are concurrent, then find the value of.
Find the equation of the straight line passing through the point (2, 1) and through the point of intersecction of the lines x + 2y = 3 and 2x – 3y = 4.
If 3a + 2b + 6c = 0 the family of straight lines ax + by + c = 0 passes through a fixed point. Find the coordinates of fixed point.