Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Distance of the point $(1,2,3)$ from the origin is $d_1=\sqrt{1^2+2^2+3^2}=\sqrt{14}$

Distance of the point $(1,2,3)$ from the plane $2x-3y+z+k=0$ is $d_2=\frac{|2(1)-3(2)+1(3)+k|}{\sqrt{2^2+(-3{)}^2+1^2}}=\frac{|k-1|}{\sqrt{14}}$

Given that $d_1\times d_2=7$

$\sqrt{14}\times \frac{|k-1|}{\sqrt{14}}=7$

$|k-1|=7$

$k-1=7$ or $k-1=-7$

$k=8$ or $k=-6$

From the given options, $k=8$.

Answer: $8$