Advertisement Remove all ads

If A, B, C Are in A.P., Prove that the Straight Lines Ax + 2y + 1 = 0, Bx + 3y + 1 = 0 and Cx + 4y + 1 = 0 Are Concurrent. - Mathematics

Answer in Brief

If a, b, c are in A.P., prove that the straight lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent.

Advertisement Remove all ads

Solution

The given lines can be written as follows:
ax + 2y + 1 = 0           ... (1)
bx + 3y + 1 = 0           ... (2)
cx + 4y + 1 = 0           ... (3)
Consider the following determinant.

\[\begin{vmatrix}a & 2 & 1 \\ b & 3 & 1 \\ c & 4 & 1\end{vmatrix}\]

Applying the transformation \[R_1 \to R_1 - R_2 \text { and } R_2 \to R_2 - R_3\],

\[\begin{vmatrix}a & 2 & 1 \\ b & 3 & 1 \\ c & 4 & 1\end{vmatrix} = \begin{vmatrix}a - b & - 1 & 0 \\ b - c & - 1 & 0 \\ c & 4 & 1\end{vmatrix}\] 

\[\Rightarrow \begin{vmatrix}a & 2 & 1 \\ b & 3 & 1 \\ c & 4 & 1\end{vmatrix} = \left( - a + b + b - c \right) = 2b - a - c\]

Given:
2b = a + c

\[\begin{vmatrix}a & 2 & 1 \\ b & 3 & 1 \\ c & 4 & 1\end{vmatrix} = a + c - a - c = 0\]

Hence, the given lines are concurrent, provided 2b = a + c.

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.11 | Q 7 | Page 83
Advertisement Remove all ads

Video TutorialsVIEW ALL [1]

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×