To find the image of the line given by the equations \((x-1)/3=(y-2)/4=(z-3)/5\) on the plane defined by \(x-2y+z+2=0\), we can follow these steps:
### Step 1: Identify the Direction Ratios of the Line and the Plane
The line can be expressed in parametric form as:
- \(x = 1 + 3t\)
- \(y = 2 + 4t\)
- \(z = 3 + 5t\)
From this, we can identify the direction ratios of the line as \( (3, 4, 5) \).
The equation of the plane \(x - 2y + z + 2 = 0\) can be rewritten as:
- \(1x - 2y + 1z + 2 = 0\)
From this, we can identify the direction ratios of the normal to the plane as \( (1, -2, 1) \).
### Step 2: Check if the Line is Parallel to the Plane
To check if the line is parallel to the plane, we calculate the dot product of the direction ratios of the line and the normal to the plane:
\[
3 \cdot 1 + 4 \cdot (-2) + 5 \cdot 1 = 3 - 8 + 5 = 0
\]
Since the dot product is zero, the line is indeed parallel to the plane.
### Step 3: Find the Image of a Point on the Line
To find the image of the line, we first need to find a point on the line. We can take \(t = 0\) to find the point \(A\):
- \(A(1, 2, 3)\)
Next, we use the formula for the image of a point with respect to a plane. The formula is given by:
\[
\text{Image} = P - 2 \frac{(P \cdot N + D)}{N \cdot N} N
\]
where \(P\) is the point, \(N\) is the normal vector of the plane, and \(D\) is the constant term from the plane's equation.
Here, \(P = (1, 2, 3)\), \(N = (1, -2, 1)\), and \(D = 2\).
### Step 4: Calculate the Image
First, calculate \(P \cdot N\):
\[
P \cdot N = 1 \cdot 1 + 2 \cdot (-2) + 3 \cdot 1 = 1 - 4 + 3 = 0
\]
Next, calculate \(N \cdot N\):
\[
N \cdot N = 1^2 + (-2)^2 + 1^2 = 1 + 4 + 1 = 6
\]
Now, substitute these values into the image formula:
\[
\text{Image} = (1, 2, 3) - 2 \cdot \frac{(0 + 2)}{6} (1, -2, 1)
\]
\[
= (1, 2, 3) - \frac{4}{6} (1, -2, 1)
\]
\[
= (1, 2, 3) - \frac{2}{3} (1, -2, 1)
\]
\[
= (1 - \frac{2}{3}, 2 + \frac{4}{3}, 3 - \frac{2}{3})
\]
\[
= \left(\frac{1}{3}, \frac{10}{3}, \frac{7}{3}\right)
\]
### Step 5: Write the Equation of the Image Line
The direction ratios of the image line will be the same as the original line, which are \( (3, 4, 5) \). Therefore, the equation of the image line can be written as:
\[
\frac{x - \frac{1}{3}}{3} = \frac{y - \frac{10}{3}}{4} = \frac{z - \frac{7}{3}}{5}
\]
### Final Answer
The image of the line on the plane is given by:
\[
\frac{x - \frac{1}{3}}{3} = \frac{y - \frac{10}{3}}{4} = \frac{z - \frac{7}{3}}{5}
\]
