A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle `|z-c|=a`, the equation of a sphere of radius is `|r-c|=a`, where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at `(-g, -f, -h)` is `x^2+y^2+z^2+2gx+2fy+2hz+c=0` and its radius is `sqrt(f^2+g^2+h^2-c)`. Q. Radius of the sphere, with `(2, -3, 4) and (-5, 6, -7)` as xtremities of a diameter, is

Options

A. (a)`sqrt((251)/(2))`
B. (b)`sqrt((251)/(3))`
C. (c)`sqrt((251)/(4))`
D. (d)`sqrt((251)/(5))`

Answer

To find the radius of the sphere given the extremities of a diameter at points \( A(2, -3, 4) \) and \( B(-5, 6, -7) \), we will follow these steps:

### Step 1: Calculate the distance \( AB \)
The distance \( AB \) between the two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) can be calculated using the distance formula in three-dimensional space:

\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Substituting the coordinates of points \( A \) and \( B \):

\[ AB = \sqrt{((-5) - 2)^2 + (6 - (-3))^2 + ((-7) - 4)^2} \]

### Step 2: Simplify the calculations
Calculating each component:

1. \( x_2 - x_1 = -5 - 2 = -7 \) so \( (-7)^2 = 49 \)
2. \( y_2 - y_1 = 6 - (-3) = 6 + 3 = 9 \) so \( 9^2 = 81 \)
3. \( z_2 - z_1 = -7 - 4 = -11 \) so \( (-11)^2 = 121 \)

Now substituting these values back into the distance formula:

\[ AB = \sqrt{49 + 81 + 121} \]

### Step 3: Calculate the sum
Calculating the sum:

\[ 49 + 81 + 121 = 251 \]

So,

\[ AB = \sqrt{251} \]

### Step 4: Calculate the radius \( r \)
Since \( AB \) is the diameter of the sphere, the radius \( r \) is half of the diameter:

\[ r = \frac{AB}{2} = \frac{\sqrt{251}}{2} \]

### Final Answer
The radius of the sphere is

\[ \frac{\sqrt{251}}{2} \]