The distance between two points P and Q is d and the length of their projections of PQ on the co-ordinate planes are `d _(1),d_(2), d_(3).` Then `d _(1) ^(2) + d_(2)^(2) + d_(3) ^(2) = kd ^(2),` where k is `"_____."`

Answer

To solve the problem, we need to find the value of \( k \) such that:

\[ d_1^2 + d_2^2 + d_3^2 = k d^2 \]

where \( d \) is the distance between two points \( P \) and \( Q \), and \( d_1, d_2, d_3 \) are the lengths of the projections of \( PQ \) on the coordinate planes.

### Step 1: Understand the projections
Let \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) be the coordinates of points \( P \) and \( Q \). The distance \( d \) between these two points is given by:

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

### Step 2: Define the projections
The projections of the line segment \( PQ \) on the coordinate planes are defined as follows:

- \( d_1 \): Projection on the XY-plane
- \( d_2 \): Projection on the XZ-plane
- \( d_3 \): Projection on the YZ-plane

The lengths of these projections can be expressed as:

\[ d_1 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ d_2 = \sqrt{(x_2 - x_1)^2 + (z_2 - z_1)^2} \]
\[ d_3 = \sqrt{(y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 3: Square the projections
Next, we square each of the projections:

\[ d_1^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \]
\[ d_2^2 = (x_2 - x_1)^2 + (z_2 - z_1)^2 \]
\[ d_3^2 = (y_2 - y_1)^2 + (z_2 - z_1)^2 \]

### Step 4: Add the squared projections
Now, we add these squared projections:

\[ d_1^2 + d_2^2 + d_3^2 = \left[(x_2 - x_1)^2 + (y_2 - y_1)^2\right] + \left[(x_2 - x_1)^2 + (z_2 - z_1)^2\right] + \left[(y_2 - y_1)^2 + (z_2 - z_1)^2\right] \]

Combining like terms, we get:

\[ d_1^2 + d_2^2 + d_3^2 = 2(x_2 - x_1)^2 + 2(y_2 - y_1)^2 + 2(z_2 - z_1)^2 \]

### Step 5: Factor out the common term
Factoring out the 2 gives us:

\[ d_1^2 + d_2^2 + d_3^2 = 2\left[(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2\right] \]

### Step 6: Relate to the distance \( d \)
We know that:

\[ d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 \]

Thus, we can substitute \( d^2 \) into our equation:

\[ d_1^2 + d_2^2 + d_3^2 = 2d^2 \]

### Step 7: Identify \( k \)
From the equation \( d_1^2 + d_2^2 + d_3^2 = k d^2 \), we can see that:

\[ k = 2 \]

### Final Answer
Thus, the value of \( k \) is:

\[ \boxed{2} \]