Question

The equation of the plane which has the property that the point $Q(5,4,5)$ is the reflection of point $P(1,2,3)$ through that plane, is $ax+by+cz=d$ where $a,b,c,d\in N$. Find the least value of $(a+b+c+d)$.

Answer 1

Answer: 17

Here midpoint is $M\equiv (3,3,4)$
and normal vector of the plane is parallel to $\overrightarrow{PQ}$
Hence $\vec{n}=4\hat{i}+2\hat{j}+2\hat{k}=2(2\hat{i}+\hat{j}+\hat{k})$

$\therefore$ Equation of plane passing through the midpoint $M$ of $PQ$, is
$2(x-3)+1(y-3)+1(z-4)=0$
$2x+y+z=13\equiv ax+by+cz=d$
$\Rightarrow a=2,b=1,c=1,d=13$
$\Rightarrow a+b+c+d=17$