The coordinate of the point of intersection of the line `(x-1)/(1)=(y+2)/(3)=(z-2)/(-2)` with the plane

Options

A. (5,10,6)
B. (10,5,6)
C. (5,5,-6)
D. (5,10,-6)

Explanation

To find the coordinates of the point of intersection of the line given by the equation

\[ \frac{x-1}{1} = \frac{y+2}{3} = \frac{z-2}{-2} \]

with the plane defined by the equation

\[ 3x + 4y + 5z - 25 = 0, \]

we will follow these steps:

### Step 1: Parameterize the line
Let us set the common ratio equal to a parameter \( \lambda \):
\[ \frac{x-1}{1} = \frac{y+2}{3} = \frac{z-2}{-2} = \lambda. \]

From this, we can express \( x \), \( y \), and \( z \) in terms of \( \lambda \):
- For \( x \):
\[ x - 1 = \lambda \implies x = \lambda + 1. \]
- For \( y \):
\[ y + 2 = 3\lambda \implies y = 3\lambda - 2. \]
- For \( z \):
\[ z - 2 = -2\lambda \implies z = -2\lambda + 2. \]

### Step 2: Substitute into the plane equation
Now, we substitute \( x \), \( y \), and \( z \) into the plane equation:
\[ 3x + 4y + 5z - 25 = 0. \]
Substituting the expressions we found:
\[ 3(\lambda + 1) + 4(3\lambda - 2) + 5(-2\lambda + 2) - 25 = 0. \]

### Step 3: Simplify the equation
Now, we simplify the left-hand side:
\[ 3\lambda + 3 + 12\lambda - 8 - 10\lambda + 10 - 25 = 0. \]
Combining like terms:
\[ (3\lambda + 12\lambda - 10\lambda) + (3 - 8 + 10 - 25) = 0, \]
which simplifies to:
\[ 5\lambda - 20 = 0. \]

### Step 4: Solve for \( \lambda \)
Now, we solve for \( \lambda \):
\[ 5\lambda = 20 \implies \lambda = 4. \]

### Step 5: Find the coordinates
Now that we have \( \lambda \), we can find the coordinates \( x \), \( y \), and \( z \):
- For \( x \):
\[ x = \lambda + 1 = 4 + 1 = 5. \]
- For \( y \):
\[ y = 3\lambda - 2 = 3(4) - 2 = 12 - 2 = 10. \]
- For \( z \):
\[ z = -2\lambda + 2 = -2(4) + 2 = -8 + 2 = -6. \]

### Final Answer
Thus, the coordinates of the point of intersection are:
\[ (5, 10, -6). \]

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