Find the co-ordinates of that point at which the line joining the points `(-2,1,4)` and `(2,0,3)` meets the `yz-`plane.

Answer

To find the coordinates of the point where the line joining the points \((-2, 1, 4)\) and \((2, 0, 3)\) meets the \(yz\)-plane, we can follow these steps:

### Step 1: Identify the Points
Let the points be:
- \( P(-2, 1, 4) \)
- \( Q(2, 0, 3) \)

### Step 2: Find the Direction Ratios
The direction ratios of the line joining points \(P\) and \(Q\) can be calculated as follows:
- The direction ratio along \(x\) is \(2 - (-2) = 4\)
- The direction ratio along \(y\) is \(0 - 1 = -1\)
- The direction ratio along \(z\) is \(3 - 4 = -1\)

Thus, the direction ratios are \( (4, -1, -1) \).

### Step 3: Write the Parametric Equations of the Line
Using the parametric form, we can express the coordinates of any point on the line as:
\[ x = -2 + 4t \]
\[ y = 1 - t \]
\[ z = 4 - t \]
where \(t\) is a parameter.

### Step 4: Determine the Condition for the \(yz\)-Plane
The \(yz\)-plane is defined by the equation \(x = 0\). We need to find the value of \(t\) when \(x = 0\):
\[ -2 + 4t = 0 \]

### Step 5: Solve for \(t\)
Solving the equation:
\[ 4t = 2 \implies t = \frac{2}{4} = \frac{1}{2} \]

### Step 6: Substitute \(t\) Back into the Parametric Equations
Now we substitute \(t = \frac{1}{2}\) into the equations for \(y\) and \(z\):
\[ y = 1 - \frac{1}{2} = \frac{1}{2} \]
\[ z = 4 - \frac{1}{2} = \frac{7}{2} \]

### Step 7: Write the Coordinates of the Point
Thus, the coordinates of the point where the line meets the \(yz\)-plane are:
\[ (0, \frac{1}{2}, \frac{7}{2}) \]

### Final Answer
The coordinates of the point at which the line joining the points \((-2, 1, 4)\) and \((2, 0, 3)\) meets the \(yz\)-plane are:
\[ \boxed{(0, \frac{1}{2}, \frac{7}{2})} \]