The ratio in which the line segment joining the points `(-2,4,5) and (3,5,-4)` is divided by the yz-plane is

Options

A. `3:2`
B. `2:3`
C. `-2:3`
D. `4:-3`

Answer

To solve the problem of finding the ratio in which the line segment joining the points \((-2, 4, 5)\) and \((3, 5, -4)\) is divided by the yz-plane, we can follow these steps:

### Step 1: Understand the yz-plane
The yz-plane is defined by the equation \(x = 0\). This means we need to find the point on the line segment where the x-coordinate is zero.

### Step 2: Use the section formula
Let the points be \(A(-2, 4, 5)\) and \(B(3, 5, -4)\). We will assume that the point \(P\) divides the line segment \(AB\) in the ratio \(k:1\). According to the section formula, the coordinates of point \(P\) can be given as:
\[ P = \left(\frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1}\right) \]
where \(A(x_1, y_1, z_1) = (-2, 4, 5)\) and \(B(x_2, y_2, z_2) = (3, 5, -4)\).

### Step 3: Set up the equation for x-coordinate
Since we want the x-coordinate of point \(P\) to be zero (as it lies on the yz-plane), we set up the equation:
\[ \frac{k \cdot 3 + 1 \cdot (-2)}{k + 1} = 0 \]

### Step 4: Solve for k
Now, we can solve this equation:
\[ k \cdot 3 - 2 = 0 \]
\[ 3k = 2 \]
\[ k = \frac{2}{3} \]

### Step 5: Write the ratio
The ratio in which the point divides the segment is \(k:1\), which is \(\frac{2}{3}:1\) or simply \(2:3\).

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

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