Key Concepts and Formulas

This problem requires the application of three fundamental concepts in 3D coordinate geometry:

1. Equation of a Sphere and its Center: The general equation of a sphere is given by ${x^2} + {y^2} + {z^2} + 2ux + 2vy + 2wz + d = 0$. The coordinates of its center, $C$, are found by taking half of the coefficients of $x, y, z$ respectively and negating them: $C = (-u, -v, -w)$.
2. Midpoint Formula: For any two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in 3D space, the coordinates of the midpoint $M$ of the line segment joining them are given by the average of their respective coordinates: $M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right)$.
3. Condition for a Point to Lie on a Plane: If a plane defined by the equation $Ax + By + Cz + D = 0$ passes through a point $P(x_0, y_0, z_0)$, then the coordinates of that point must satisfy the plane's equation. That is, substituting $x_0, y_0, z_0$ into the equation must result in a true statement: $Ax_0 + By_0 + Cz_0 + D = 0$.

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Step-by-Step Solution

Step 1: Determine the Centers of the Given Spheres

The problem involves the "midpoint of the line joining the centres of the spheres," so our initial step is to find the coordinates of these centers from their given equations.

• Sphere 1: ${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13$ To find the center, we compare this equation with the general form ${x^2} + {y^2} + {z^2} + 2ux + 2vy + 2wz + d = 0$. By comparing the coefficients of $x, y, z$: $2u = 6 \implies u = 3$ $2v = -8 \implies v = -4$ $2w = -2 \implies w = -1$ Therefore, the center of the first sphere, $C_1$, is $(-u, -v, -w) = (-3, -(-4), -(-1)) = (-3, 4, 1)$.

• Sphere 2: ${x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8$ Similarly, comparing with the general form of a sphere: $2u = -10 \implies u = -5$ $2v = 4 \implies v = 2$ $2w = -2 \implies w = -1$ Therefore, the center of the second sphere, $C_2$, is $(-u, -v, -w) = (-(-5), -(2), -(-1)) = (5, -2, 1)$.

Step 2: Calculate the Midpoint of the Line Segment Joining the Centers

The problem states that the given plane passes through the midpoint of the line segment connecting $C_1$ and $C_2$. We will use the midpoint formula with the coordinates of $C_1 = (-3, 4, 1)$ and $C_2 = (5, -2, 1)$.

Let $C_1 = (x_1, y_1, z_1)$ and $C_2 = (x_2, y_2, z_2)$. The midpoint $M = (x_M, y_M, z_M)$ is calculated as: $x_M = \frac{x_1+x_2}{2} = \frac{-3+5}{2} = \frac{2}{2} = 1$ $y_M = \frac{y_1+y_2}{2} = \frac{4+(-2)}{2} = \frac{2}{2} = 1$ $z_M = \frac{z_1+z_2}{2} = \frac{1+1}{2} = \frac{2}{2} = 1$ So, the midpoint of the line joining the centers is $M = (1, 1, 1)$.

Step 3: Substitute the Midpoint Coordinates into the Plane Equation

We are given the equation of the plane as $2ax - 3ay + 4az + 6 = 0$. Since the plane passes through the midpoint $M(1, 1, 1)$, the coordinates of $M$ must satisfy the plane's equation. This allows us to form an equation in terms of the unknown 'a'.

Substitute $x=1$, $y=1$, and $z=1$ into the plane equation: $2a(1) - 3a(1) + 4a(1) + 6 = 0$ $2a - 3a + 4a + 6 = 0$

Step 4: Solve for 'a'

Now, we simplify the algebraic equation from Step 3 to find the value of 'a'. Combine the terms involving 'a': $(2 - 3 + 4)a + 6 = 0$ $(3)a + 6 = 0$ Subtract 6 from both sides: $3a = -6$ Divide by 3: $a = \frac{-6}{3}$ $a = -2$

Thus, the value of 'a' is -2.

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Common Mistakes & Tips

• Sign Errors in Center Coordinates: A frequent mistake is forgetting to negate $u, v, w$ when determining the center of the sphere. Always remember that the center is $(-u, -v, -w)$.
• Incorrect Coefficient Matching: Ensure you correctly identify $2u, 2v, 2w$ from the sphere equation. For instance, if a term is just $x$, then $2u=1$, so $u=1/2$.
• Arithmetic Precision: Be careful with additions, subtractions, and divisions, especially with negative numbers, during the midpoint calculation.
• Substitution Accuracy: Double-check that you substitute the correct $x, y, z$ values into the corresponding variables in the plane equation.

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Summary

This problem is a straightforward application of fundamental 3D geometry concepts. The solution involved three main steps: first, extracting the centers of the two given spheres from their standard equations; second, calculating the midpoint of the line segment connecting these two centers using the midpoint formula; and finally, utilizing the condition that this midpoint lies on the given plane by substituting its coordinates into the plane's equation to solve for the unknown parameter 'a'. By systematically applying these principles, we found the value of 'a' to be -2.

The final answer is $\boxed{\text{-2}}$, which corresponds to option (C).