$If the lines a_{1} x + b_{1} y + 1 = 0, a_{2} x + b_{2} y + 1 = 0 and a_{3} x + b_{3} y + 1 = 0 are concurrent, then the points (a_{1}, b_{1}), (a_{2}, b_{2}) and (a_{3}, b_{3}) will be collinear .$

True

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False

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Solution

The correct option is A True
$If the lines a_{1} x + b_{1} y + 1 = 0, a_{2} x + b_{2} y + 1 = 0 and a_{3} x + b_{3} y + 1 = 0 are concurrent Then, ⎛ ⎜ ⎝ \begin{matrix} a_{1} & b_{1} & 1 a_{2} & b_{2} & 1 a_{3} & b_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0 This is the required condition for the concurrence of three straight lines . Also we can say that the triangle formed using the points (a_{1}, b_{1}), (a_{2}, b_{2}), and (a_{3}, b_{3}) is zero area . Thus, the points are collinear .$

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Similar questions

If the lines a_{1} x + b_{1} y + 1 = 0, a_{2} x + b_{2} y + 1 = 0 and a_{3} x + b_{3} y + 1 = 0 are concurrent, then the points (a_{1}, b_{1}), (a_{2}, b_{2}) and (a_{3}, b_{3}) will be collinear .

(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.

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

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

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

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

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If the lines a1x+b1y+1=0,a2x+b2y+1=0and a3x+b3y+1=0 are concurrent, then the points (a1,b1),(a2,b2) and (a3,b3) will be collinear.

$If the lines a_{1} x + b_{1} y + 1 = 0, a_{2} x + b_{2} y + 1 = 0 and a_{3} x + b_{3} y + 1 = 0 are concurrent, then the points (a_{1}, b_{1}), (a_{2}, b_{2}) and (a_{3}, b_{3}) will be collinear .$