Which of the statement is true? The coordinate planes divide the line joining the points ` (4,7,-2) and (-5,8,3)`:

Options

A. All externally
B. two externally and one internally
C. two internally and one externally
D. None of the above

Answer

To determine how the coordinate planes divide the line segment joining the points \( A(4, 7, -2) \) and \( B(-5, 8, 3) \), we will analyze the intersection of the line segment with the three coordinate planes: the XY-plane, the YZ-plane, and the ZX-plane.

### Step 1: Finding the intersection with the XY-plane
The XY-plane is defined by \( z = 0 \). To find the point of intersection, we use the section formula.

Let the point \( C \) divide the line segment \( AB \) in the ratio \( k:1 \). The coordinates of point \( C \) can be expressed as:
\[ C = \left( \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} \right) \]
where \( A(4, 7, -2) \) and \( B(-5, 8, 3) \).

Setting \( z = 0 \):
\[ 0 = \frac{k \cdot 3 + 1 \cdot (-2)}{k + 1} \]
This simplifies to:
\[ 0 = 3k - 2 \implies 3k = 2 \implies k = \frac{2}{3} \]
Thus, the XY-plane divides the line segment in the ratio \( 2:3 \) internally.

### Step 2: Finding the intersection with the YZ-plane
The YZ-plane is defined by \( x = 0 \). Let the point \( D \) divide the line segment \( AB \) in the ratio \( p:1 \).

Setting \( x = 0 \):
\[ 0 = \frac{p \cdot (-5) + 1 \cdot 4}{p + 1} \]
This simplifies to:
\[ 0 = -5p + 4 \implies 5p = 4 \implies p = \frac{4}{5} \]
Thus, the YZ-plane divides the line segment in the ratio \( 4:5 \) internally.

### Step 3: Finding the intersection with the ZX-plane
The ZX-plane is defined by \( y = 0 \). Let the point \( M \) divide the line segment \( AB \) in the ratio \( m:1 \).

Setting \( y = 0 \):
\[ 0 = \frac{m \cdot 8 + 1 \cdot 7}{m + 1} \]
This simplifies to:
\[ 0 = 8m + 7 \implies 8m = -7 \implies m = -\frac{7}{8} \]
Thus, the ZX-plane divides the line segment in the ratio \( 7:8 \) externally.

### Conclusion
- The XY-plane divides the segment in the ratio \( 2:3 \) (internally).
- The YZ-plane divides the segment in the ratio \( 4:5 \) (internally).
- The ZX-plane divides the segment in the ratio \( 7:8 \) (externally).

### Final Answer
The correct statement is that two planes (XY and YZ) divide the line segment internally, while one plane (ZX) divides it externally.