Find the ratio in which the line joining the points (2, 4, 5), (3,5,-4) is divided by the yz-plane.

Answer

To find the ratio in which the line joining the points \( A(2, 4, 5) \) and \( B(3, 5, -4) \) is divided by the YZ-plane, we follow these steps:

### Step 1: Understand the YZ-plane
The YZ-plane is defined by the equation \( x = 0 \). This means that any point on the YZ-plane has its x-coordinate equal to 0.

### Step 2: Identify the coordinates of points A and B
We have:
- Point A: \( (2, 4, 5) \)
- Point B: \( (3, 5, -4) \)

### Step 3: Set up the section formula
Let the point that divides the line segment AB in the ratio \( M:N \) be \( P(0, y, z) \) since it lies on the YZ-plane. According to the section formula, the x-coordinate of point P can be expressed as:

\[ x = \frac{x_2 \cdot M + x_1 \cdot N}{M + N} \]

where \( (x_1, y_1, z_1) = (2, 4, 5) \) and \( (x_2, y_2, z_2) = (3, 5, -4) \).

### Step 4: Substitute the coordinates into the formula
Since \( x = 0 \) for the YZ-plane, we have:

\[ 0 = \frac{3M + 2N}{M + N} \]

### Step 5: Solve the equation
To solve for the ratio \( M:N \), we can multiply both sides by \( M + N \):

\[ 0 = 3M + 2N \]

Rearranging gives:

\[ 3M + 2N = 0 \implies 3M = -2N \implies \frac{M}{N} = -\frac{2}{3} \]

### Step 6: Interpret the ratio
This means that the ratio \( M:N \) is \( 2:3 \), but since the ratio is negative, it indicates that the point divides the segment externally.

### Final Answer
Thus, the ratio in which the line joining the points \( (2, 4, 5) \) and \( (3, 5, -4) \) is divided by the YZ-plane is \( 2:3 \) (external division).

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