Find the coordinates of the point where the line through `(3, 4, 1) and (5, 1, 6)` crosses XY-plane.

Answer

To find the coordinates of the point where the line through the points \( A(3, 4, 1) \) and \( B(5, 1, 6) \) crosses the XY-plane, we can follow these steps:

### Step 1: Identify the Points
Let point \( A \) be \( (3, 4, 1) \) and point \( B \) be \( (5, 1, 6) \).

### Step 2: Determine the Parametric Equations of the Line
The line segment joining points \( A \) and \( B \) can be expressed in parametric form. If \( C \) is a point on the line that divides \( AB \) in the ratio \( \lambda : 1 \), the coordinates of point \( C \) can be given by:

\[ C = \left( \frac{\lambda \cdot 5 + 1 \cdot 3}{\lambda + 1}, \frac{\lambda \cdot 1 + 1 \cdot 4}{\lambda + 1}, \frac{\lambda \cdot 6 + 1 \cdot 1}{\lambda + 1} \right) \]

### Step 3: Set the Z-coordinate to Zero
Since point \( C \) lies on the XY-plane, its z-coordinate must be zero. Therefore, we set the z-coordinate equation to zero:

\[ \frac{\lambda \cdot 6 + 1 \cdot 1}{\lambda + 1} = 0 \]

### Step 4: Solve for \( \lambda \)
To solve for \( \lambda \), we can cross-multiply:

\[ \lambda \cdot 6 + 1 = 0 \implies 6\lambda = -1 \implies \lambda = -\frac{1}{6} \]

### Step 5: Substitute \( \lambda \) Back into the Parametric Equations
Now we substitute \( \lambda = -\frac{1}{6} \) back into the equations for the coordinates of point \( C \):

1. For the x-coordinate:
\[ x = \frac{-\frac{1}{6} \cdot 5 + 3}{-\frac{1}{6} + 1} = \frac{-\frac{5}{6} + 3}{\frac{5}{6}} = \frac{-\frac{5}{6} + \frac{18}{6}}{\frac{5}{6}} = \frac{\frac{13}{6}}{\frac{5}{6}} = \frac{13}{5} \]

2. For the y-coordinate:
\[ y = \frac{-\frac{1}{6} \cdot 1 + 4}{-\frac{1}{6} + 1} = \frac{-\frac{1}{6} + 4}{\frac{5}{6}} = \frac{-\frac{1}{6} + \frac{24}{6}}{\frac{5}{6}} = \frac{\frac{23}{6}}{\frac{5}{6}} = \frac{23}{5} \]

### Step 6: Write the Final Coordinates
Thus, the coordinates of the point where the line crosses the XY-plane are:

\[ C\left(\frac{13}{5}, \frac{23}{5}, 0\right) \]

### Summary of the Solution
The coordinates of the point where the line through \( (3, 4, 1) \) and \( (5, 1, 6) \) crosses the XY-plane are \( \left(\frac{13}{5}, \frac{23}{5}, 0\right) \).