A point on the line `(x-1)/1=(y-3)/2=(z+2)/7` has coordinates

Options

A. `(3,5,4)`
B. `(2,5,5)`
C. `(-1,-1,5)`
D. `(2,-1,0)`

Answer

To find a point on the line given by the equation \((x-1)/1 = (y-3)/2 = (z+2)/7\), we can follow these steps:

### Step 1: Understand the Line Equation
The equation \((x-1)/1 = (y-3)/2 = (z+2)/7\) represents a line in 3D space. We can express this in parametric form by introducing a parameter \(t\):
- \(x = 1 + t\)
- \(y = 3 + 2t\)
- \(z = -2 + 7t\)

### Step 2: Choose a Value for \(t\)
To find specific coordinates on the line, we can choose a value for \(t\). Let's choose \(t = 1\).

### Step 3: Substitute \(t\) into the Parametric Equations
Now we will substitute \(t = 1\) into the parametric equations:
- For \(x\):
\[ x = 1 + 1 = 2 \]
- For \(y\):
\[ y = 3 + 2(1) = 3 + 2 = 5 \]
- For \(z\):
\[ z = -2 + 7(1) = -2 + 7 = 5 \]

### Step 4: Write the Coordinates
Thus, the coordinates of the point on the line when \(t = 1\) are:
\[ (2, 5, 5) \]

### Step 5: Verify the Coordinates
To ensure that the point \((2, 5, 5)\) lies on the line, we substitute these coordinates back into the original line equation:
- Check:
\[ \frac{2-1}{1} = \frac{5-3}{2} = \frac{5+2}{7} \]
- Calculate:
\[ \frac{1}{1} = \frac{2}{2} = \frac{7}{7} \]
- This simplifies to:
\[ 1 = 1 = 1 \]
Since all parts are equal, the point \((2, 5, 5)\) does indeed lie on the line.

### Conclusion
The coordinates of a point on the line are \((2, 5, 5)\).

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