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IfO, Pare the points $(0,0,0),(2,3,-1)$ respectively, then what is the equation to the plane through $\mathrm{P}$ at right angles to $\mathrm{OP}$ ?
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Verified Answer
The correct answer is:
$2 x+3 y-z=14$
Since, coordinates of points $\mathrm{O}$ and $\mathrm{P}$ are $(0,0,0)$ and $(2,3,-1)$ respectively. Direction ratios of $\mathrm{OP}$ are $ < 2,3,-1\rangle$
The plane is perpendicular to OP. So, its equation is $2 x+3 y-z+d=0$
Since, this plane passes through $(2,3,-1) ; 2 \times 2+3 \times$
$3-1 \times-1+d=0$
$\Rightarrow 4+9+1+\mathrm{d}=0$
$\Rightarrow \mathrm{d}=-14$
On putting the value of $\mathrm{d}$ in Eq. (i) $2 x+3 y-z-14=0$
$\Rightarrow 2 x+3 y-z=14$
which is required equation of plane
The plane is perpendicular to OP. So, its equation is $2 x+3 y-z+d=0$
Since, this plane passes through $(2,3,-1) ; 2 \times 2+3 \times$
$3-1 \times-1+d=0$
$\Rightarrow 4+9+1+\mathrm{d}=0$
$\Rightarrow \mathrm{d}=-14$
On putting the value of $\mathrm{d}$ in Eq. (i) $2 x+3 y-z-14=0$
$\Rightarrow 2 x+3 y-z=14$
which is required equation of plane
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