Find the equation of line through the origin and a point where line `(x-1)/2=(y-2)/4=(z-3)/4`. Cuts
the z-axis.

Answer

To find the equation of the line through the origin and the point where the given line intersects the z-axis, we can follow these steps:

### Step 1: Determine the point of intersection with the z-axis
The given line is represented by the equation:
\[ \frac{x - 1}{2} = \frac{y - 2}{4} = \frac{z - 3}{4} \]
A point on the z-axis has coordinates of the form (0, 0, z). To find the point where the given line intersects the z-axis, we set \(x = 0\) and \(y = 0\).

### Step 2: Substitute x and y into the equation
Substituting \(x = 0\) and \(y = 0\) into the equation:
\[ \frac{0 - 1}{2} = \frac{0 - 2}{4} = \frac{z - 3}{4} \]
This gives us:
\[ \frac{-1}{2} = \frac{-2}{4} = \frac{z - 3}{4} \]

### Step 3: Simplify the equations
From \(\frac{-1}{2} = \frac{-2}{4}\), we can see that both sides are equal, so we can now focus on finding \(z\):
\[ \frac{-1}{2} = \frac{z - 3}{4} \]

### Step 4: Cross-multiply to solve for z
Cross-multiplying gives:
\[ -1 \cdot 4 = -2 \cdot (z - 3) \]
This simplifies to:
\[ -4 = -2z + 6 \]

### Step 5: Rearrange the equation to solve for z
Rearranging the equation:
\[ -2z = -4 - 6 \]
\[ -2z = -10 \]
\[ z = 5 \]

### Step 6: Identify the point of intersection
Thus, the point of intersection \(P\) on the z-axis is:
\[ P(0, 0, 5) \]

### Step 7: Find the direction vector of the line OP
The line passes through the origin \(O(0, 0, 0)\) and the point \(P(0, 0, 5)\). The direction vector \(\vec{OP}\) is:
\[ \vec{OP} = P - O = (0 - 0, 0 - 0, 5 - 0) = (0, 0, 5) \]

### Step 8: Write the equation of the line
The equation of the line in symmetric form is given by:
\[ \frac{x - 0}{0} = \frac{y - 0}{0} = \frac{z - 0}{5} \]
This can be simplified to:
\[ \frac{x}{0} = \frac{y}{0} = \frac{z}{5} \]

### Final Equation of the Line
Thus, the required equation of the line through the origin and the point where the given line intersects the z-axis is:
\[ x = 0, \quad y = 0, \quad z = 5t \quad (t \in \mathbb{R}) \]