find the coordinate of a point where `A(3, 4, 1 )` and `B(5, 1, 6)` connecting line intersect the plane XY

Options

A. `(13/5, 23/5, 0)`
B. (0, 0, 0)
C. `(-13/5, -23/5, 0)`
D. `(-13/5, 23/5, 0)`

Answer

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

### Step 1: Determine the Direction Ratios
The direction ratios of the line connecting points A and B can be calculated using the coordinates of these points.

- Coordinates of A: \( A(3, 4, 1) \)
- Coordinates of B: \( B(5, 1, 6) \)

The direction ratios (d) can be calculated as follows:
\[ d_x = x_B - x_A = 5 - 3 = 2 \]
\[ d_y = y_B - y_A = 1 - 4 = -3 \]
\[ d_z = z_B - z_A = 6 - 1 = 5 \]

Thus, the direction ratios are \( (2, -3, 5) \).

### Step 2: Write the Parametric Equations of the Line
Using point A and the direction ratios, we can write the parametric equations of the line:

\[ \frac{x - 3}{2} = \frac{y - 4}{-3} = \frac{z - 1}{5} = t \]

From this, we can express \( x, y, z \) in terms of \( t \):
\[ x = 3 + 2t \]
\[ y = 4 - 3t \]
\[ z = 1 + 5t \]

### Step 3: Find the Intersection with the XY Plane
The XY plane is defined by the equation \( z = 0 \). To find the intersection, we set \( z = 0 \) in the equation for \( z \):

\[ 1 + 5t = 0 \]

Solving for \( t \):
\[ 5t = -1 \implies t = -\frac{1}{5} \]

### Step 4: Substitute \( t \) Back to Find \( x \) and \( y \)
Now we substitute \( t = -\frac{1}{5} \) back into the equations for \( x \) and \( y \):

For \( x \):
\[ x = 3 + 2\left(-\frac{1}{5}\right) = 3 - \frac{2}{5} = \frac{15}{5} - \frac{2}{5} = \frac{13}{5} \]

For \( y \):
\[ y = 4 - 3\left(-\frac{1}{5}\right) = 4 + \frac{3}{5} = \frac{20}{5} + \frac{3}{5} = \frac{23}{5} \]

### Step 5: Write the Coordinates of the Intersection Point
Thus, the coordinates of the point where the line intersects the XY plane are:
\[ \left( \frac{13}{5}, \frac{23}{5}, 0 \right) \]

### Final Answer
The coordinates of the intersection point are:
\[ \left( \frac{13}{5}, \frac{23}{5}, 0 \right) \]