Let the plane ax+by+cz+d=0 bisect the line joining the points (4,−3,1) and (2,3,−5) at right angles. If a,b,c,d are integers, then the minimum value of (a2+b2+c2+d2) is:
Answer: 28
Solutions · 1
Official solution
Midpoint M=(3,0,−2); direction PQ=(2,−6,6)∝(1,−3,3) (the plane's normal).
\[1(x-3)-3(y-0)+3(z+2)=0 \implies x-3y+3z+3=0.\]
\[a=1,\ b=-3,\ c=3,\ d=3 \implies a^2+b^2+c^2+d^2=1+9+9+9=28.\]