Q4 of 319 Page 24

If the lines p1x + q1y = 1, p2x + q2y = 1 and p3x + q3y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.

Given:


p1x + q1y = 1


p2x + q2y = 1


p3x + q3y = 1


To prove:


The points (p1, q1), (p2, q2) and (p3, q3) are collinear.


Concept Used:


If three lines are concurrent then determinant of equation is zero.


Explanation:


The given lines can be written as follows:


p1 x + q1 y – 1 = 0 … (1)


p2 x + q2 y – 1 = 0 … (2)


p3 x + q3 y – 1 = 0 … (3)


It is given that the three lines are concurrent.


∴


⇒


⇒
Hence proved, This is the condition for the collinearity of the three points, (p1, q1), (p2, q2) and (p3, q3).


More from this chapter

All 319 →
2

For what value of λ are the three lines 2x – 5y + 3 = 0, 5x – 9y + λ = 0 and x – 2y + 1 = 0 concurrent?

 3

Find the conditions that the straight lines y = m1x + c1, y = m2x + c2 and y = m3x + c3 may meet in a point.

 5

Show that the straight lines L1 = (b + c)x + ay + 1 = 0, L2 = (c + a)x + by + 1 = 0 and L3 = (a + b)x + cy + 1 = 0 are concurrent.

 6

If the three lines ax + a2y + 1 = 0, bx + b2y + 1 = 0 and cx + c2y + 1 = 0 are concurrent, show that at least two of three constant a, b, c are equal.