Question

Find the value of $(m + n)$ where, m and n are comparatively prime then find the radius of the sphere which is in the form of $\dfrac{m}{n}$. Given that: The sphere inscribed in the tetrahedron where the vertices are as follows;- $A \equiv (6,0,0)$,$B \equiv (0,4,0)$,$C \equiv (0,0,2)$ and $D \equiv (0,0,0)$ respectively.
(a) $3$
(b) $10$
(c) $0$
(d) $5$

Answer 1

Hint: As given in the question, the sphere is inscribed in the tetrahedron, so the length of the perpendicular from the centre upon each of the vertices of the tetrahedron implies the radius of the sphere. Then by using the distance formula of a point from the plane the desired answer is obtained.

Complete step-by-step answer:

Let us consider the vector having the magnitude and direction to the sphere which is inscribed in a tetrahedron.
Let us assume that the tetrahedron ABDC is a plane.
As a result of the radius of the sphere ‘r’, the sphere is located at the centre ‘O’ inside the tetrahedron at (r,r,r) respectively because the problem consists of three-dimensional geometry.
Hence, the distance between the plane ABDC and the centre ‘O’ can be found as,
$\dfrac{{(O - G).P}}{{|P|}}$
Where, G is any point on the plane ABDC and P is any vector perpendicular to the plane ABDC.
As a result, vector P perpendicular to the plane ABDC is found as,
$V = (A - C) \times (B - C)$
$ \Rightarrow V = (8,12,24)$ …….($\because $using cross-product of vector identity)
Thus, from the distance formula we can write
\[ \Rightarrow \dfrac{{(O - G).P}}{{|P|}} = - r\]
Since, here we have adjusted the radius ‘r’ for the sake of easiness,
Hence, substituting the respective values in the above formula, we get
$ \Rightarrow \dfrac{{\left[ {(r,r,r) - (0,0,2)} \right].P}}{{|P|}} = - r$
Now, using the concept of dot product and magnitude P, we get to know that
\[ \Rightarrow \dfrac{{\left( {r,r,r - 2} \right).(8,12,24)}}{{(8,12,24)}} = - r\]
Simplifying the equation, we get
$ \Rightarrow \dfrac{{44r - 48}}{{28}} = - r$
Taking denominator to RHS, we get
$
   \Rightarrow 44r - 48 = - 28r \\
   \Rightarrow 44r + 28r = 48 = 72r \;
 $
Hence,
$r = \dfrac{{48}}{{72}}$
That to be written in the simplest form as,
$r = \dfrac{2}{3} = \dfrac{m}{n}$ ….(dividing numerator and denominator by 24)
Hence, we get the values of m and n as $2$ and $3$ respectively as prime numbers!
Hence, proved!
As a result, $m + n = 2 + 3$
$m + n = 5$
$\therefore $Therefore, it implies that the correct option is (d).
So, the correct answer is “Option d”.

Note: We note that the vertices are given in the form of coordinates as it implies vector geometry seems to be the respective points at each of the vertex point of tetrahedron respectively say, A, B, C, D. One must clarify the vector concept especially both cross and dot product so that the solution becomes easy.