The position vectors of the four angular points of a tetrahedron $OABC$ are $(0,0,0);(0,0,2);(0,4,0)$ and $(6,0,0)$ respectively.A point $P$ inside the tetrahedron is at the same distance $r$ from the four plane faces of the tetrahedron.Find the value of $r.$

The equation of a plane passing through three points $(x_1,y_1,z_1),(x_2,y_2,z_2)$ and $(x_3,y_3,z_3)$ is $\begin{vmatrix}x-x_1&y-y_1&z-z_1\\x_2-x_1&y_2-y_1&z_2-z_1\\x_3-x_1&y_3-y_1&z_3-z_1\end{vmatrix}=0$

Using this,i found the equations of the planes $OAB,OBC,OAC$ are $x=0,z=0,y=0$ respectively and the equation of the plane $ABC$ is $2x+3y+6z-12=0$.Let the coordinates of $P$ be $(x_1,y_1,z_1)$.

Distance of $P$ from $OAB$ plane is $x_1$.
Distance of $P$ from $OBC$ plane is $z_1$.
Distance of $P$ from $OAC$ plane is $y_1$.
Distance of $P$ from $ABC$ plane is $\frac{2x_1+3y_1+6z_1-12}{7}$.

According to the question,
I solved the equations to find $x_1=y_1=z_1=3$.So $r=3$
But the answer given in the book is $\frac{2}{3}$.I dont know where have i gone wrong?Please help me.Thanks.

  • $\begingroup$ Obviously, P is the incenter of the tetrahedron. $\endgroup$ – Lucian Nov 8 '15 at 19:11

The distance of $P$ from $ABC$ plane is $$\frac{|2x_1+3y_1+6z_1-12|}{7}.$$

So, solving $$x_1=y_1=z_1=\frac{|2x_1+3y_1+6z_1-12|}{7}=r$$ gives $$(x_1,y_1,z_1,r)=(3,3,3,3),\left(\frac 23,\frac 23,\frac 23,\frac 23\right).$$

But the former is outside the tetrahedron.

| cite | improve this answer | |
  • $\begingroup$ How do we say that $(3,3,3)$ is outside the tetrahedron?,Sir.Is there any method for checking this. $\endgroup$ – Vinod Kumar Punia Nov 8 '15 at 16:31
  • $\begingroup$ @VinodKumarPunia: Let $F(x,y,z)=2x+3y+6z−12$. We have $F(0,0,0)=−12\lt 0$. This means that we have to have at least $F(x,y,z)\lt 0$ in order for $(x,y,z)$ to be inside the tetrahedron. Now, $F(3,3,3)=21\gt 0$, which means that $(3,3,3)$ is outside the tetrahedron. $\endgroup$ – mathlove Nov 8 '15 at 17:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.