If the plane 2ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining centres of the spheres `x^(2)+y^(2)+z^(2)-2x-4y+2z-3=0` and `x^(2)+y^(2)+z^(2)+x-y-2z=13` then a equals

Options

A. `-1`
B. 1
C. 24/13
D. 2

Answer

To solve the problem, we need to find the value of \( a \) such that the plane \( 2ax - 3ay + 4az + 6 = 0 \) passes through the midpoint of the line joining the centers of the two given spheres.

### Step 1: Find the centers of the spheres

The equations of the spheres are:

1. \( x^2 + y^2 + z^2 - 2x - 4y + 2z - 3 = 0 \)
2. \( x^2 + y^2 + z^2 + x - y - 2z = 13 \)

To find the center of a sphere given by the equation \( x^2 + y^2 + z^2 + 2ax + 2by + 2cz + d = 0 \), the center is given by the coordinates \( (-a, -b, -c) \).

For the first sphere:
- Rearranging gives us:
\[ x^2 - 2x + y^2 - 4y + z^2 + 2z - 3 = 0 \]
- The center \( C_1 \) is:
\[ C_1 = \left( \frac{2}{2}, \frac{4}{2}, \frac{-2}{2} \right) = (1, 2, -1) \]

For the second sphere:
- Rearranging gives us:
\[ x^2 + y^2 + z^2 + x - y - 2z - 13 = 0 \]
- The center \( C_2 \) is:
\[ C_2 = \left( -\frac{1}{2}, \frac{1}{2}, \frac{2}{2} \right) = \left( -\frac{1}{2}, \frac{1}{2}, 1 \right) \]

### Step 2: Find the midpoint of the line joining the centers

The midpoint \( M \) of the line segment joining points \( C_1 \) and \( C_2 \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting the coordinates of \( C_1 \) and \( C_2 \):
\[ M = \left( \frac{1 + \left(-\frac{1}{2}\right)}{2}, \frac{2 + \frac{1}{2}}{2}, \frac{-1 + 1}{2} \right) \]
Calculating each coordinate:
- \( x \): \( \frac{1 - \frac{1}{2}}{2} = \frac{\frac{1}{2}}{2} = \frac{1}{4} \)
- \( y \): \( \frac{2 + \frac{1}{2}}{2} = \frac{\frac{5}{2}}{2} = \frac{5}{4} \)
- \( z \): \( \frac{-1 + 1}{2} = 0 \)

Thus, the midpoint \( M \) is \( \left( \frac{1}{4}, \frac{5}{4}, 0 \right) \).

### Step 3: Substitute the midpoint into the plane equation

The plane equation is:
\[ 2ax - 3ay + 4az + 6 = 0 \]
Substituting \( M \):
\[ 2a\left(\frac{1}{4}\right) - 3a\left(\frac{5}{4}\right) + 4a(0) + 6 = 0 \]
This simplifies to:
\[ \frac{2a}{4} - \frac{15a}{4} + 6 = 0 \]
Combining terms:
\[ -\frac{13a}{4} + 6 = 0 \]
Rearranging gives:
\[ -\frac{13a}{4} = -6 \]
Multiplying both sides by -4:
\[ 13a = 24 \]
Thus:
\[ a = \frac{24}{13} \]

### Final Answer
The value of \( a \) is \( \frac{24}{13} \).