Question

The equation of the sphere on the join of $2\widehat i + 2\widehat j - 3\widehat k\;,\;5\widehat i - \widehat j + 2\widehat k$ as a diameter is
(a)${x^2} + {y^2} + {z^2} - 7x - y + z + 2 = 0$
(b)${x^2} + {y^2} + {z^2} + 7x + y - z - 2 = 0$
(c)${x^2} + {y^2} + {z^2} - 7x + y + z - 2 = 0$
(d)${x^2} + {y^2} + {z^2} + 7x + y + z - 2 = 0$

Answer 1

Hint:- We need to find out the centre and the radius of the sphere from the given data in the question. We can then use the formula of a sphere whose centre and radius is given.

Complete step-by-step answer:
It has been given that the diameter of the sphere is the join of the two points $2\widehat i + 2\widehat j - 3\widehat k\;,\;5\widehat i - \widehat j + 2\widehat k$ which can written in cartesian coordinate form as $\left( {2,2, - 3} \right),\left( {5, - 1,2} \right)$
We can alternatively say that these two points are the endpoints of the given diameter.
We know that the centre of the line joining the points $\left( {x,y,z} \right)$ and $\left( {a,b,c} \right)$ is $\left( {\dfrac{{x + a}}{2},\dfrac{{y + b}}{2},\dfrac{{z + c}}{2}} \right) \equiv \left( {\dfrac{{\left( {x\hat i + y\hat j + z\hat k} \right) + \left( {a\hat i + b\hat j + c\hat k} \right)}}{2}} \right)$
Now using the above formula, we can find of the center of sphere
Thus, the centre is given by
\[\dfrac{{\left( {2\widehat i + 2\widehat j - 3\widehat k} \right) + \left( {5\widehat i - \widehat j + 2\widehat k} \right)}}{2}\]
Further simplifying and separating out the terms, we get the center of the sphere as \[\dfrac{7}{2}\widehat i + \dfrac{1}{2}\widehat j - \dfrac{1}{2}\widehat k\]
We know that the length of the line joining the two points $\left( {x,y,z} \right)$ and $\left( {a,b,c} \right)$ is given by \[\sqrt {{{\left( {x - a} \right)}^2} + {{\left( {y - b} \right)}^2} + {{\left( {z - c} \right)}^2}} \]
Hence, using this formula we can calculate the length of the diameter joining the points $\left( {2,2, - 3} \right),\left( {5, - 1,2} \right)$ is $\sqrt {{{\left( {2 - 5} \right)}^2} + {{\left( {2 + 1} \right)}^2} + {{\left( { - 3 - 2} \right)}^2}} $
We know the length of the radius of the sphere is half the length of the diameter.
Thus, the length of the radius is
$\dfrac{{\sqrt {{{\left( {2 - 5} \right)}^2} + {{\left( {2 + 1} \right)}^2} + {{\left( { - 3 - 2} \right)}^2}} }}{2} = \dfrac{{\sqrt {43} }}{2}$
Now, let us have a look at the formula for the equation of sphere with center $a\widehat i + b\widehat j + c\widehat k$ and radius r is given by
${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} + {\left( {z - c} \right)^2} = {r^2}$
Hence, the equation of sphere is given by
${\left( {x - \dfrac{7}{2}} \right)^2} + {\left( {y - \dfrac{1}{2}} \right)^2} + {\left( {z + \dfrac{1}{2}} \right)^2} = {\left( {\dfrac{{\sqrt {43} }}{2}} \right)^2}$
$ \Rightarrow {x^2} - 7x + \dfrac{{49}}{4} + {y^2} - y + \dfrac{1}{4} + {z^2} + z + \dfrac{1}{4} = \dfrac{{43}}{4}$
Multiplying by 4 throughout, we get
$ \Rightarrow 4{x^2} - 28x + 49 + 4{y^2} - 4y + 1 + 4{z^2} + 4z + 1 = 43$
We can now rearrange the term to get
$ \Rightarrow 4{x^2} - 28x + 4{y^2} - 4y + 4{z^2} + 4z + 8 = 0$
$ \Rightarrow {x^2} - 7x + {y^2} - y + {z^2} + z + 2 = 0$
$ \Rightarrow {x^2} + {y^2} + {z^2} - 7x - y + z + 2 = 0$

Hence the correct answer is (a)

Note:- In such types of questions, where the vector form of coordinates is given, it can be easily solved by converting it into the cartesian coordinates. Then we can use simple geometry and find out the center and radius of the sphere in the same way we find that out for a 2-D circle.