If the line joining the points A(-1,2,5) and B(3,4,-10) intersects the xy-plane at the point (x,y,z), then y/x is equal to ______

Answer

To solve the problem, we need to find the point where the line joining points A(-1, 2, 5) and B(3, 4, -10) intersects the xy-plane. The intersection with the xy-plane occurs when the z-coordinate is 0.

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B**:
- Point A: \( A(-1, 2, 5) \)
- Point B: \( B(3, 4, -10) \)

2. **Determine the parametric equations of the line joining A and B**:
The parametric equations can be derived using the formula for the coordinates of a point dividing the line segment in the ratio \( m:n \):
\[ x = x_1 + t(x_2 - x_1) \]
\[ y = y_1 + t(y_2 - y_1) \]
\[ z = z_1 + t(z_2 - z_1) \]
Here, \( t \) is a parameter that varies from 0 to 1.

Substituting the coordinates:
\[ x = -1 + t(3 - (-1)) = -1 + 4t \]
\[ y = 2 + t(4 - 2) = 2 + 2t \]
\[ z = 5 + t(-10 - 5) = 5 - 15t \]

3. **Set the z-coordinate to 0 to find the intersection with the xy-plane**:
\[ 5 - 15t = 0 \]
Solving for \( t \):
\[ 15t = 5 \implies t = \frac{1}{3} \]

4. **Substitute \( t \) back into the equations for \( x \) and \( y \)**:
- For \( x \):
\[ x = -1 + 4\left(\frac{1}{3}\right) = -1 + \frac{4}{3} = \frac{-3 + 4}{3} = \frac{1}{3} \]
- For \( y \):
\[ y = 2 + 2\left(\frac{1}{3}\right) = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \]

5. **Calculate \( \frac{y}{x} \)**:
\[ \frac{y}{x} = \frac{\frac{8}{3}}{\frac{1}{3}} = 8 \]

Thus, the value of \( \frac{y}{x} \) is **8**.