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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class11  >>  3-D Geometry
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Q)

If the position vector of the points $A\:and\:B$ are $\hat i-\hat j+3\hat k$ and $3\hat i+3\hat j+3\hat k$ and the equation of a plane is $\overrightarrow r.(5\hat i+2\hat j-7\hat k)+9=0$, then the points $A\:and\:B$ are?

$\begin{array}{1 1} (a)\:lie\: on\: the\: plane.\:\qquad\:(b)\:are\: on\: the\: same\:side\:of\:the\:plane.\:\qquad\:(c)\:are\:on\:opp.\:sides\:of\:the\:plane.\:\qquad\:(d)\:none\:of\:these \end{array} $

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A)
If the point satisfies the equation of the plane then it lies on the plane.
If the points when substituted in the equation of the plane are of same sign, then
they lie on the same side of the plane.
If they are of opp. signs, then they lie on opp. sides of the plane.
Here the equation of the plane in cartesian form is $5x+2y-7z+9=0$ and
the points are $A(1,-1,3)\:\:and\:\:B(3,3,3)$
Substituting the coordinates of $A$ and $B$ on the equation of the plane,
$A$ gives $ -ve$ sign and $B$ gives $+ve$ sign.
$\therefore$ They lie on opposite sides of the plane.
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