Question
Chord $$AB$$ is a diameter of the sphere $$\left| {\overrightarrow r - 2\overrightarrow i - \overrightarrow j + 6\overrightarrow k } \right| = \sqrt {18} .$$ If the coordinates of $$A$$ are $$\left( {3,\,2,\, - 2} \right),$$ then the coordinates of $$B$$ are :
A.
$$\left( {1,\,0,\,10} \right)$$
B.
$$\left( {1,\,0,\, - 10} \right)$$
C.
$$\left( { - 1,\,0,\,10} \right)$$
D.
None of these
Answer :
$$\left( {1,\,0,\, - 10} \right)$$
Solution :
The equation of the sphere is
$$\left| {\overrightarrow r - 2\overrightarrow i - \overrightarrow j + 6\overrightarrow k } \right| = \sqrt {18} $$
$$ \Rightarrow $$ Its centre is at the point $$\left( {\overrightarrow r - 2\overrightarrow i - \overrightarrow j + 6\overrightarrow k } \right),$$ i.e., at $$\left( {2,\,1,\, - 6} \right).$$
Coordinates of $$A$$ are $$\left( {3,\,2,\, - 2} \right).$$
Let the coordinates of $$B$$ be $$\left( {\alpha ,\,\beta ,\,\gamma } \right).$$
$$\eqalign{
& {\text{Then, }}\frac{{3 + \alpha }}{2} = 2,\,\frac{{2 + \beta }}{2} = 1{\text{ and }}\frac{{ - 2 + \lambda }}{2} = - 6 \cr
& \Rightarrow \alpha = 1,\,\beta = 0,\,\gamma = - 10 \cr} $$
Therefore, coordinates of $$B$$ are $$\left( {1,\,0,\, - 10} \right).$$