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If the equation of the plane bisecting the line segment joining the points $P(3,2,4)$ and $Q(-1,0,-2)$ and perpendicular to $P Q$ is $a x+b y+c z+d=0$, then $a c+b d$
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Mid-point of line segment joining the points $P(3,2,4)$ and $Q(-1,0,-2)$ is $R(1,1,1)$ and direction ratios of line segment $P Q$ is $4,2,6$, so direction ratios of normal to the plane is
$$
\langle a, b, c\rangle=\langle 4,2,6\rangle .
$$
So, equation of plane will be
$$
4 x+2 y+6 z+d=0
$$
Since, plane (i) bisect the line segment joining $P Q$.
So, $\quad d=-12$
Therefore, $\quad a c+b d=0$.
$$
\langle a, b, c\rangle=\langle 4,2,6\rangle .
$$
So, equation of plane will be
$$
4 x+2 y+6 z+d=0
$$
Since, plane (i) bisect the line segment joining $P Q$.
So, $\quad d=-12$
Therefore, $\quad a c+b d=0$.
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