The condition for three lines $a_{1}$ x+ $b_{1}$ y+ $c_{1}$ = 0, $a_{2}$ x+ $b_{2}$ y+ $c_{2}$ = 0 and $a_{3}$ x+ $b_{3}$ y+ $c_{3}$ = 0 , to be concurrent is

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

True

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False

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Solution

The correct option is A
True

This is a result from determinants.

$a_{1}$ x+ $b_{1}$ y+ $c_{1}$ = 0

$a_{2}$ x+ $b_{2}$ y+ $c_{2}$ = 0 and

$a_{3}$ x+ $b_{3}$ y+ $c_{3}$ = 0 are concurrent if

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

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The condition for three lines $a_{1} x + b_{1} y + c_{1} = 0$ , $a_{2} x + b_{2} y + c_{2} = 0$ and $a_{3} x + b_{3} y + c_{3} = 0$ , to be concurrent is

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

The condition for three lines $a_{1}$ x+ $b_{1}$ y+ $c_{1}$ = 0, $a_{2}$ x+ $b_{2}$ y+ $c_{2}$ = 0 and $a_{3}$ x+ $b_{3}$ y+ $c_{3}$ = 0 , to be concurrent is

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

(A);

(a_{1} x + b_{1} y + c_{1}) + (a_{2} x + b_{2} y + c_{2}) + a_{3} x + b_{3} y + c_{3}) = 0,

then the lines

a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0, a_{3} x + b_{3} y + c_{3} = 0

can not be parallel.
(R): If sum of three straight lines is identically

0,

then they are either concurrent or parallel.

Three lines given by $L_{1} : a_{1} x + b_{1} y + c_{1} = 0, L_{2} : a_{2} x + b_{2} y + c_{2} = 0 and L_{3} : a_{3} x + b_{3} y + c_{3} = 0$ are always concurrent given that

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

Three lines given by $4 : a_{1} x + b_{1} y + c = 0, L_{2} : a_{2} x + b_{2} y + c_{2} = 0 a n d L_{3} : a_{3} x + b_{3} y + c_{3} = 0$ are always concurrent given that

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$ =0

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The condition for three lines a1x+b1y+c1 = 0, a2x+b2y+c2 = 0 and a3x+b3y+c3 = 0 , to be concurrent is ∣∣ ∣∣a1b1c1a2b2c2a3b3c3∣∣ ∣∣=0

The correct option is A True This is a result from determinants. a1x+b1y+c1 = 0 a2x+b2y+c2 = 0 and a3x+b3y+c3 = 0 are concurrent if ∣∣ ∣∣a1b1c1a2b2c2a3b3c3∣∣ ∣∣=0

The condition for three lines $a_{1}$ x+ $b_{1}$ y+ $c_{1}$ = 0, $a_{2}$ x+ $b_{2}$ y+ $c_{2}$ = 0 and $a_{3}$ x+ $b_{3}$ y+ $c_{3}$ = 0 , to be concurrent is

$∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$