## Question

If the lines *ax* + *y* + 1 = 0, *x* + *by* + 1 = 0 and *x* + *y* + *c* = 0 (*a*, *b* and *c*being distinct and difference from 1) are concurrent, then find the value of.

### Solution

1

The given lines are concurrent, then

(applying C_{2} → C_{2} – C_{1} and C_{3 }→ C_{3} – C_{1})

Expanding along first row

Dividing by (1 – *a*)(1 – *b*)(1 – *c*) then

Hence,

.

#### SIMILAR QUESTIONS

Are the points (2, 1) and (–3, 5) on the same or opposite side of the line 3*x* – 2*y* + 1 = 0?

Is the point (2, –7) lie on origin side of the line 2*x* + *y *+ 2 = 0?

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

3*x* – 4*y* + 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 4*x* + 3*y* – 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 4*p*^{2} + *p*’^{2}.

Find the distance between the lines 5*x* – 12*y* + 2 = 0 and

5*x* – 12*y* – 3 = 0.

Find the equations of the line parallel to 5*x* – 12*y* + 26 = 0 and at a distance of 4 units from it.

Find the equation of the straight line passing through the point (2, 1) and through the point of intersecction of the lines *x* + 2*y* = 3 and 2*x* – 3*y* = 4.