The coordinates of the foot of the perpendicular drawn from the origin to the line joining the point `(-9,4,5)` and `(10 ,0,-1)` will be

Options

A. (-3,2,1)
B. (1,2,2,)
C. (4,5,3)
D. none of these

Explanation

To find the coordinates of the foot of the perpendicular drawn from the origin to the line joining the points \((-9, 4, 5)\) and \((10, 0, -1)\), we can follow these steps:

### Step 1: Determine the direction ratios of the line segment joining the two points.
The direction ratios can be calculated as follows:
\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
Substituting the given points:
\[ = (10 - (-9), 0 - 4, -1 - 5) = (19, -4, -6) \]

### Step 2: Write the parametric equations of the line.
Using the direction ratios and one of the points, we can write the equations of the line:
\[ \frac{x + 9}{19} = \frac{y - 4}{-4} = \frac{z - 5}{-6} = \lambda \]
From this, we can express \(x\), \(y\), and \(z\) in terms of \(\lambda\):
\[ x = 19\lambda - 9, \quad y = 4 - 4\lambda, \quad z = 5 - 6\lambda \]

### Step 3: Set up the direction ratios from the origin to the point on the line.
The coordinates of the foot of the perpendicular from the origin \(O(0, 0, 0)\) to the point \(M(x, y, z)\) on the line are:
\[ OM = (19\lambda - 9 - 0, 4 - 4\lambda - 0, 5 - 6\lambda - 0) = (19\lambda - 9, 4 - 4\lambda, 5 - 6\lambda) \]

### Step 4: Use the condition for perpendicularity.
For the vectors \(OM\) and the direction ratios of the line to be perpendicular, their dot product must equal zero:
\[ (19)(19\lambda - 9) + (-4)(4 - 4\lambda) + (-6)(5 - 6\lambda) = 0 \]
Expanding this:
\[ 361\lambda - 171 - 16 + 16\lambda - 30 + 36\lambda = 0 \]
Combining like terms:
\[ (361 + 16 + 36)\lambda - 217 = 0 \]
\[ 413\lambda - 217 = 0 \]
Thus,
\[ \lambda = \frac{217}{413} \]

### Step 5: Substitute \(\lambda\) back to find \(x\), \(y\), and \(z\).
Substituting \(\lambda\) into the equations for \(x\), \(y\), and \(z\):
\[ x = 19\left(\frac{217}{413}\right) - 9 = \frac{4123}{413} - \frac{3707}{413} = \frac{416}{413} = \frac{58}{59} \]
\[ y = 4 - 4\left(\frac{217}{413}\right) = \frac{1652}{413} - \frac{868}{413} = \frac{784}{413} = \frac{112}{59} \]
\[ z = 5 - 6\left(\frac{217}{413}\right) = \frac{2065}{413} - \frac{1302}{413} = \frac{763}{413} = \frac{109}{59} \]

### Final Result:
The coordinates of the foot of the perpendicular from the origin to the line joining the points \((-9, 4, 5)\) and \((10, 0, -1)\) are:
\[ \left(\frac{58}{59}, \frac{112}{59}, \frac{109}{59}\right) \]