A rod of length $2$ units whose one end is $(1,0,-1)$ and the other end touches the plane $x-2y+2z+4=0,$ then find the center of the region which the rod traces on the plane.
The rod sweeps out the figure which is a cone.
And in this question ,we need to find the center of the circle,which is projection of the apex point $P(1,0,-1)$ onto the plane $x-2y+2z+4=0$.
Let the center of the circle be $O(x_1,y_1,z_1)$.So the direction ratios of the line $OP$ are $(x_1-1,y_1,z_1+1)$.Since $OP$ and the normal to the plane are parallel so their direction ratios are proportional.
$\frac{x_1-1}{1}=\frac{y_1}{-2}=\frac{z_1+1}{2}=\lambda$(say)
Now $x_1=\lambda+1,y_1=-2\lambda,z_1=2\lambda-1$
Since $O$ lies on the plane $x-2y+2z+4=0$,So $x_1-2y_1+2z_1+4=0$
$\lambda+1-2(-2\lambda)+2(2\lambda-1)+4=0$
$\lambda=\frac{-1}{3}$
So the coordinates of the $O$ are $(\frac{2}{3},\frac{2}{3},\frac{-5}{3})$
But the answer given in my book is $O(\frac{4}{3},\frac{-2}{3},\frac{1}{3})$
What is wrong in my method.Please help me.Is my answer correct or the one given in my book?
Your answer is correct. The point given by the book is not even on the plane.