If the distance from the point P(1, 1, 1) to the line passing through the points `Q(0, 6, 8) andR(-1, 4, 7)` is expressed in the form `sqrt((p)/(q))`, where p and q are co-prime, then the value of `((q+p)(p+q-1))/(2)` is equal to

Options

A. `4950`
B. `5050`
C. `5150`
D. None of these

Answer

To find the distance from the point \( P(1, 1, 1) \) to the line passing through the points \( Q(0, 6, 8) \) and \( R(-1, 4, 7) \), we can use the formula for the distance \( d \) from a point to a line in 3D space. The steps are as follows:

### Step 1: Find the direction vector of the line
The direction vector \( \vec{QR} \) from point \( Q \) to point \( R \) is given by:
\[ \vec{QR} = R - Q = (-1 - 0, 4 - 6, 7 - 8) = (-1, -2, -1) \]

### Step 2: Find the vector from point \( Q \) to point \( P \)
The vector \( \vec{QP} \) from point \( Q \) to point \( P \) is:
\[ \vec{QP} = P - Q = (1 - 0, 1 - 6, 1 - 8) = (1, -5, -7) \]

### Step 3: Calculate the cross product \( \vec{QP} \times \vec{QR} \)
To find the distance, we need the magnitude of the cross product of \( \vec{QP} \) and \( \vec{QR} \):
\[ \vec{QP} \times \vec{QR} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -5 & -7 \\ -1 & -2 & -1 \end{vmatrix} \]
Calculating this determinant:
\[ = \hat{i}((-5)(-1) - (-7)(-2)) - \hat{j}((1)(-1) - (-7)(-1)) + \hat{k}((1)(-2) - (-5)(-1)) \]
\[ = \hat{i}(5 - 14) - \hat{j}(-1 + 7) + \hat{k}(-2 - 5) \]
\[ = \hat{i}(-9) - \hat{j}(6) + \hat{k}(-7) \]
\[ = (-9, -6, -7) \]

### Step 4: Find the magnitude of the cross product
The magnitude of the cross product \( |\vec{QP} \times \vec{QR}| \) is:
\[ |\vec{QP} \times \vec{QR}| = \sqrt{(-9)^2 + (-6)^2 + (-7)^2} = \sqrt{81 + 36 + 49} = \sqrt{166} \]

### Step 5: Find the magnitude of the direction vector \( |\vec{QR}| \)
The magnitude of the direction vector \( \vec{QR} \) is:
\[ |\vec{QR}| = \sqrt{(-1)^2 + (-2)^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]

### Step 6: Calculate the distance \( d \)
The distance \( d \) from point \( P \) to the line is given by:
\[ d = \frac{|\vec{QP} \times \vec{QR}|}{|\vec{QR}|} = \frac{\sqrt{166}}{\sqrt{6}} = \sqrt{\frac{166}{6}} = \sqrt{\frac{83}{3}} \]

### Step 7: Identify \( p \) and \( q \)
Here, \( p = 83 \) and \( q = 3 \). Since 83 and 3 are co-prime, we can proceed to the next calculation.

### Step 8: Calculate \( \frac{(q+p)(p+q-1)}{2} \)
Now we calculate:
\[ (q + p) = 3 + 83 = 86 \]
\[ (p + q - 1) = 83 + 3 - 1 = 85 \]
Thus,
\[ \frac{(q+p)(p+q-1)}{2} = \frac{86 \times 85}{2} = \frac{7310}{2} = 3655 \]

### Final Answer
The value of \( \frac{(q+p)(p+q-1)}{2} \) is \( 3655 \).

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