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If the lines $x^2+2 x y-35 y^2-4 x+44 y$ $-12=0$ and $5 x+\lambda y-8=0$ are concurrent, then $\lambda$ equals
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The correct answer is:
$2$
Given line,
$x^2+2 x y-35 y^2-4 x+44 y-2=0$
$\Rightarrow \quad(x+7 y-6)(x-5 y+2)=0$
$\Rightarrow x+7 y-6=0$ and $x-5 y+2=0$ and
$5 x+\lambda y-8=0$ are concurrent.
$\therefore \quad\left|\begin{array}{ccc}1 & 7 & -6 \\ 1 & -5 & 2 \\ 5 & \lambda & -8\end{array}\right|=0$
$1(40-2 \lambda)-7(-8-10)-6(1 \lambda+25)=0$
$\Rightarrow \quad 40-2 \lambda+126-6 \lambda-150=0$
$\Rightarrow \quad 8 \lambda=16$
$\Rightarrow \quad \lambda=2$
$x^2+2 x y-35 y^2-4 x+44 y-2=0$
$\Rightarrow \quad(x+7 y-6)(x-5 y+2)=0$
$\Rightarrow x+7 y-6=0$ and $x-5 y+2=0$ and
$5 x+\lambda y-8=0$ are concurrent.
$\therefore \quad\left|\begin{array}{ccc}1 & 7 & -6 \\ 1 & -5 & 2 \\ 5 & \lambda & -8\end{array}\right|=0$
$1(40-2 \lambda)-7(-8-10)-6(1 \lambda+25)=0$
$\Rightarrow \quad 40-2 \lambda+126-6 \lambda-150=0$
$\Rightarrow \quad 8 \lambda=16$
$\Rightarrow \quad \lambda=2$
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