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Let $\mathrm{A}=(-3,-2,7)$ and $\mathrm{B}=(3,1,-2)$. Let a plane perpendicular to the line segment $A B$ divide $A B$ in the ratio $2: 1$. Then the intercept made by the plane on $y$-axis is
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The correct answer is:
$-1$
Let plane $\mathrm{P}$ divides Line $\mathrm{AB}$ at $\mathrm{Q}$ in ratio 2:1
$\Rightarrow Q=(1,0,1)$
Now, equation of plane passing through $(1,0,1)$ with DRS $(6,3,-9)$ is
$\begin{aligned} & 6(x-1)+3(y-0)-9(z-1)=0 \\ & \Rightarrow 6 x+3 y-9 z+3=0 \\ & \Rightarrow-2 x-y+3 z=1 \\ & \Rightarrow \frac{x}{-\frac{1}{2}}+\frac{y}{(-1)}+\frac{z}{\left(\frac{1}{3}\right)}=1\end{aligned}$
hence y intercepts is -1
$\Rightarrow Q=(1,0,1)$
Now, equation of plane passing through $(1,0,1)$ with DRS $(6,3,-9)$ is
$\begin{aligned} & 6(x-1)+3(y-0)-9(z-1)=0 \\ & \Rightarrow 6 x+3 y-9 z+3=0 \\ & \Rightarrow-2 x-y+3 z=1 \\ & \Rightarrow \frac{x}{-\frac{1}{2}}+\frac{y}{(-1)}+\frac{z}{\left(\frac{1}{3}\right)}=1\end{aligned}$
hence y intercepts is -1
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