To find the ratio in which the y-z plane divides the line segment formed by joining the points A(-2, 4, 7) and B(3, -5, 8), we can follow these steps:
### Step 1: Understand the problem
We need to find the ratio in which the y-z plane divides the line segment joining the points A and B. The y-z plane is defined by the equation x = 0.
### Step 2: Set up the coordinates
Let the coordinates of point A be A(-2, 4, 7) and the coordinates of point B be B(3, -5, 8).
### Step 3: Use the section formula
If the y-z plane divides the line segment AB in the ratio k:1, then the coordinates of the point C (the point of division) can be expressed using the section formula:
\[
C = \left( \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} \right)
\]
where \( (x_1, y_1, z_1) \) are the coordinates of point A and \( (x_2, y_2, z_2) \) are the coordinates of point B.
### Step 4: Substitute the coordinates
Substituting the coordinates of points A and B into the section formula:
- \( x_1 = -2, y_1 = 4, z_1 = 7 \)
- \( x_2 = 3, y_2 = -5, z_2 = 8 \)
The coordinates of point C become:
\[
C = \left( \frac{k \cdot 3 + 1 \cdot (-2)}{k + 1}, \frac{k \cdot (-5) + 1 \cdot 4}{k + 1}, \frac{k \cdot 8 + 1 \cdot 7}{k + 1} \right)
\]
### Step 5: Set the x-coordinate to 0
Since point C lies on the y-z plane, we set the x-coordinate equal to 0:
\[
\frac{3k - 2}{k + 1} = 0
\]
### Step 6: Solve for k
To solve for k, we set the numerator equal to zero:
\[
3k - 2 = 0 \implies 3k = 2 \implies k = \frac{2}{3}
\]
### Step 7: Find the ratio
The ratio in which the y-z plane divides the line segment AB is k:1, which is:
\[
\frac{2}{3}:1 \implies 2:3
\]
### Final Answer
The ratio in which the y-z plane divides the line segment joining the points A and B is **2:3**.
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