The distance of point of intersection of lines `(x-4)/1=(x+3)/-4=(z-1)/7 and (x-1)/2=(y+1)/-3=(z+10)/8` from `(1,-4,7)` is

Options

A. 6
B. `sqrt(24)`
C. `sqrt(26)`
D. none of these

Answer

To find the distance of the point of intersection of the given lines from the point (1, -4, 7), we will follow these steps:

### Step 1: Parameterize the lines
The first line is given by:
\[ \frac{x - 4}{1} = \frac{y + 3}{-4} = \frac{z - 1}{7} \]
Let \( t \) be the parameter. Then we can express the coordinates as:
\[ x_1 = t + 4, \quad y_1 = -4t - 3, \quad z_1 = 7t + 1 \]

The second line is given by:
\[ \frac{x - 1}{2} = \frac{y + 1}{-3} = \frac{z + 10}{8} \]
Let \( \lambda \) be the parameter. Then we can express the coordinates as:
\[ x_2 = 2\lambda + 1, \quad y_2 = -3\lambda - 1, \quad z_2 = 8\lambda - 10 \]

### Step 2: Set the coordinates equal to find the intersection
Since we are looking for the point of intersection, we set the coordinates equal:
1. \( t + 4 = 2\lambda + 1 \)
2. \( -4t - 3 = -3\lambda - 1 \)
3. \( 7t + 1 = 8\lambda - 10 \)

### Step 3: Solve the equations
From the first equation:
\[ t + 4 = 2\lambda + 1 \implies t = 2\lambda - 3 \tag{1} \]

From the second equation:
\[ -4t - 3 = -3\lambda - 1 \implies -4(2\lambda - 3) - 3 = -3\lambda - 1 \]
Substituting \( t \) from (1):
\[ -8\lambda + 12 - 3 = -3\lambda - 1 \implies -8\lambda + 9 = -3\lambda - 1 \]
\[ -5\lambda = -10 \implies \lambda = 2 \]

Now substitute \( \lambda = 2 \) back into (1) to find \( t \):
\[ t = 2(2) - 3 = 1 \]

### Step 4: Find the coordinates of the intersection point
Substituting \( t = 1 \) into the first line's parameterization:
\[ x = 1 + 4 = 5, \quad y = -4(1) - 3 = -7, \quad z = 7(1) + 1 = 8 \]
So the intersection point is \( (5, -7, 8) \).

### Step 5: Calculate the distance from the point (1, -4, 7)
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(5 - 1)^2 + (-7 + 4)^2 + (8 - 7)^2} \]
\[ = \sqrt{(4)^2 + (-3)^2 + (1)^2} = \sqrt{16 + 9 + 1} = \sqrt{26} \]

### Final Answer
The distance of the point of intersection from the point (1, -4, 7) is \( \sqrt{26} \).

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