First slide

Equation of line in Straight lines

Question

If the lines $x + y + 1 = 0, 4 x + 3 y + 4 = 0$ and $x + αy + β = 0$ where $α^{2} + β^{2} = 2$ are concurrent then

Easy

A

$α = 1, β = - 1$
B

$α = 1, β = \pm 1$
C

$α = - 1, β = \pm 1$
D

$α = \pm 1, β = 1$

Solution

The given lines are
$\begin{array}{l} x + y + 1 = 0 \\ 4 x + 3 y + 4 = 0 \\ x + αy + β = 0 \end{array}$
If these lines are concurrent, then $|\begin{array}{lll} 1 1 1 \\ 4 3 4 \\ 1 α β \end{array}| = 0$
To solve the equation use elementary column operations
$C_{2} \to C_{2} - C_{1}, C_{3} \to C_{3} - C_{1}$
$|\begin{array}{ccc} 1 & 0 & 0 \\ 4 & - 1 & 0 \\ 1 & α - 1 & β - 1 \end{array}| = 0 \begin{array}{l} 1 (1 - β - 0) = 0 \\ β = 1 \end{array}$
Given $α^{2} + β^{2} = 2 \Rightarrow α^{2} = 1 \Rightarrow α = \pm 1$
Therefore $α = \pm 1 and β = 1$

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