If three lines whose equations are y = m 1 x + c 1 , y = m 2 x + c 2 and y = m 3 x + c 3 are concurrent, then show that m 1 (c 2 – c 3 ) + m 2 (c 3 – c 1 ) + m 3 (c 1 – c 2 ) = 0.

Question

Philoid · Accepted Answer

The equation of the given lines are

y = m₁x + c₁ ---------(1)

y = m₂x + c₂ ---------(2)

y = m₃x + c₃ ---------(3)

On subtracting equation (1) from (2), we get,

0 = (m₂ - m₁)x + (c₂ – c₁)

⇒ (m₂ - m₁)x = (c₂ – c₁)

⇒ x =

On substituting this value of x in (1), we get,

Thus,

is the point of intersection of lines (1) and (2).

It is given that lines (1), (2) and (3) are concurrent.

Thus, the point of intersection of lines (1) and (2) will also satisfy equation (3).

If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3 are concurrent, then show that m1(c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0.

If three lines whose equations are y = m₁x + c₁, y = m₂x + c₂ and y = m₃x + c₃ are concurrent, then show that m₁(c₂ – c₃) + m₂ (c₃ – c₁) + m₃ (c₁ – c₂) = 0.