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If the distance between the points $(7,1,-3)$ and $(4,5, \lambda)$ is 13 units, then what is one of the values of $\lambda$ ? $\quad$
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Verified Answer
The correct answer is:
9
We have,
$13=\sqrt{(4-7)^{2}+(5-1)^{2}+(\lambda+3)^{2}}$
$169=9+16+\lambda^{2}+9+6 \lambda$
$\Rightarrow \quad \lambda^{2}+6 \lambda-135=0$
$\Rightarrow \quad \lambda^{2}+15 \lambda-9 \lambda-135=0$
$\Rightarrow \quad \lambda(\lambda+15)-9(\lambda+15)=0$
$\Rightarrow(\lambda+15)(\lambda-9)=0$
$\Rightarrow \quad \lambda=-15$ or $\lambda=9$
$13=\sqrt{(4-7)^{2}+(5-1)^{2}+(\lambda+3)^{2}}$
$169=9+16+\lambda^{2}+9+6 \lambda$
$\Rightarrow \quad \lambda^{2}+6 \lambda-135=0$
$\Rightarrow \quad \lambda^{2}+15 \lambda-9 \lambda-135=0$
$\Rightarrow \quad \lambda(\lambda+15)-9(\lambda+15)=0$
$\Rightarrow(\lambda+15)(\lambda-9)=0$
$\Rightarrow \quad \lambda=-15$ or $\lambda=9$
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