Question
If $$\left( {2,\,3,\,5} \right)$$ is one end of a diameter of the sphere $${x^2} + {y^2} + {z^2} - 6x - 12y - 2z + 20 = 0,$$ then the coordinates of the other end of the diameter are :
A.
$$\left( {4,\,3,\,5} \right)$$
B.
$$\left( {4,\,3,\, - 3} \right)$$
C.
$$\left( {4,\,9,\, - 3} \right)$$
D.
$$\left( {4,\, - 3,\,3} \right)$$
Answer :
$$\left( {4,\,9,\, - 3} \right)$$
Solution :
For given sphere centre is $$\left( {3,\,6,\,1} \right)$$
Coordinates of one end of diameter of the sphere are $$\left( {2,\,3,\,5} \right).$$
Let the coordinates of the other end of diameter are $$\left( {\alpha ,\,\beta ,\,\gamma } \right)$$
$$\eqalign{
& \therefore \,\frac{{\alpha + 2}}{2} = 3,\,\,\frac{{\beta + 3}}{2} = 6,\,\,\frac{{\gamma + 5}}{2} = 1 \cr
& \Rightarrow \alpha = 4,\,\beta = 9{\text{ and }}\,\gamma = - 3 \cr} $$
$$\therefore $$ Coordinate of other end of diameter are $$\left( {4,\,9,\, - 3} \right)$$