If the plane `2ax-3ay+4az+6=0` passes through the mid point of the line joining the centre of the spheres `x^(2)+y^(2)+z^(2)+6x-8y-2z=13 and x^(2)+y^(2)+z^(2)-10x+4y-2z=8`, then `alpha` equals

Options

A. `2`
B. `-2`
C. `1`
D. `-1`

Answer

To solve the given problem, we need to find the value of \( \alpha \) such that the plane \( 2\alpha x - 3\alpha y + 4\alpha z + 6 = 0 \) passes through the midpoint of the line joining the centers of the two spheres given by the equations:

1. \( x^2 + y^2 + z^2 + 6x - 8y - 2z = 13 \)
2. \( x^2 + y^2 + z^2 - 10x + 4y - 2z = 8 \)

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

The general equation of a sphere is given by:

\[ x^2 + y^2 + z^2 + 2Ax + 2By + 2Cz + D = 0 \]

From this, we can identify the center of the sphere as \( (-A, -B, -C) \).

**For the first sphere:**

The equation is:

\[ x^2 + y^2 + z^2 + 6x - 8y - 2z - 13 = 0 \]

Here, \( 2A = 6 \), \( 2B = -8 \), and \( 2C = -2 \).

Thus, the center is:

\[ \left(-\frac{6}{2}, -\frac{-8}{2}, -\frac{-2}{2}\right) = (-3, 4, 1) \]

**For the second sphere:**

The equation is:

\[ x^2 + y^2 + z^2 - 10x + 4y - 2z - 8 = 0 \]

Here, \( 2A = -10 \), \( 2B = 4 \), and \( 2C = -2 \).

Thus, the center is:

\[ \left(-\frac{-10}{2}, -\frac{4}{2}, -\frac{-2}{2}\right) = (5, -2, 1) \]

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

The midpoint \( M \) of the line joining the points \( (-3, 4, 1) \) and \( (5, -2, 1) \) is given by:

\[ M = \left( \frac{-3 + 5}{2}, \frac{4 - 2}{2}, \frac{1 + 1}{2} \right) = \left( \frac{2}{2}, \frac{2}{2}, \frac{2}{2} \right) = (1, 1, 1) \]

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

The plane equation is:

\[ 2\alpha x - 3\alpha y + 4\alpha z + 6 = 0 \]

Substituting \( (1, 1, 1) \):

\[ 2\alpha(1) - 3\alpha(1) + 4\alpha(1) + 6 = 0 \]

This simplifies to:

\[ 2\alpha - 3\alpha + 4\alpha + 6 = 0 \]

Combining like terms:

\[ (2 - 3 + 4)\alpha + 6 = 0 \]

This simplifies to:

\[ 3\alpha + 6 = 0 \]

### Step 4: Solve for \( \alpha \)

Now, we can solve for \( \alpha \):

\[ 3\alpha = -6 \]

\[ \alpha = -2 \]

### Final Answer

Thus, the value of \( \alpha \) is:

\[ \alpha = -2 \]