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If the lines $3 y+4 x=1, y=x+5$ and $5 y+b x=3$ are ooncurrent, then what is the value of $b ?$
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Verified Answer
The correct answer is:
6
The equation of given lines are
$3 y+4 x=1$ ...(i)
$y=x+5$ ...(ii)
and $5 y+b x=3$ ...(iii)
On solving Eqs. (i) and (ii), we get $x=-2$ and $y=3$
If these lines are concurrent, then these values must satisfy the third equation $15-2 b=3 \Rightarrow 2 b=12 \Rightarrow b=6$
$ALTERNATESOLUTION$:
Given equation of lines are $3 y+4 x-1=0$
$x+5-y=0$ and
$b x+5 y-3=0$
Since, the given lines are concurrent $\therefore$ The value of determinant made by coeff of equations is $0 .$
ie, $\left|\begin{array}{ccc}4 & 3 & -1 \\ 1 & -1 & 5 \\ b & 5 & -3\end{array}\right|=0$
$\Rightarrow 4(3-25)-3(-3-5 b)-1(5+b)=0$
$\Rightarrow-88+4+14 b=0$
$\Rightarrow-84=-14 b$
$\Rightarrow b=6 .$
$3 y+4 x=1$ ...(i)
$y=x+5$ ...(ii)
and $5 y+b x=3$ ...(iii)
On solving Eqs. (i) and (ii), we get $x=-2$ and $y=3$
If these lines are concurrent, then these values must satisfy the third equation $15-2 b=3 \Rightarrow 2 b=12 \Rightarrow b=6$
$ALTERNATESOLUTION$:
Given equation of lines are $3 y+4 x-1=0$
$x+5-y=0$ and
$b x+5 y-3=0$
Since, the given lines are concurrent $\therefore$ The value of determinant made by coeff of equations is $0 .$
ie, $\left|\begin{array}{ccc}4 & 3 & -1 \\ 1 & -1 & 5 \\ b & 5 & -3\end{array}\right|=0$
$\Rightarrow 4(3-25)-3(-3-5 b)-1(5+b)=0$
$\Rightarrow-88+4+14 b=0$
$\Rightarrow-84=-14 b$
$\Rightarrow b=6 .$
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