The shorteast distance from `(1, 1, 1)` to the line of intersection of the pair of planes `xy+yz+zx+y^2=0` is

Options

A. `sqrt((8)/(7))`
B. `(2)/(sqrt(3))`
C. `(1)/(sqrt(3))`
D. `(2)/(3)`

Answer

To find the shortest distance from the point \( P(1, 1, 1) \) to the line of intersection of the pair of planes given by the equation \( xy + yz + zx + y^2 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation of the planes
We start with the equation of the planes:
\[ xy + yz + zx + y^2 = 0 \]
We can rearrange this equation as:
\[ xy + y^2 + yz + zx = 0 \]
Factoring out \( y \) from the first two terms and \( z \) from the last two terms gives:
\[ y(x + y) + z(x + y) = 0 \]
Now, we can factor out \( (x + y) \):
\[ (x + y)(y + z) = 0 \]

### Step 2: Determine the equations of the planes
From the factored equation, we have two cases:
1. \( x + y = 0 \) which implies \( x = -y \)
2. \( y + z = 0 \) which implies \( y = -z \)

### Step 3: Find the line of intersection
From the equations \( x = -y \) and \( y = -z \), we can express all coordinates in terms of a single parameter \( t \):
- Let \( y = t \)
- Then \( x = -t \)
- And \( z = -y = -t \)

Thus, the parametric equations of the line of intersection can be written as:
\[ (x, y, z) = (-t, t, -t) \]

### Step 4: Find the direction ratios of the line
The direction ratios of the line can be derived from the parametric equations:
\[ \text{Direction ratios} = (1, -1, 1) \]

### Step 5: Calculate the shortest distance from point to line
The formula for the shortest distance \( d \) from a point \( P(x_1, y_1, z_1) \) to a line given by a point \( A(x_0, y_0, z_0) \) and direction ratios \( (a, b, c) \) is:
\[ d = \frac{|(P - A) \cdot (b, c, d)|}{\sqrt{a^2 + b^2 + c^2}} \]
Here, we can take point \( A \) on the line when \( t = 0 \):
\[ A(0, 0, 0) \]
And point \( P(1, 1, 1) \).

### Step 6: Calculate the vector \( P - A \)
The vector \( P - A \) is:
\[ P - A = (1 - 0, 1 - 0, 1 - 0) = (1, 1, 1) \]

### Step 7: Calculate the dot product
Now we calculate the dot product:
\[ (1, 1, 1) \cdot (1, -1, 1) = 1 \cdot 1 + 1 \cdot (-1) + 1 \cdot 1 = 1 - 1 + 1 = 1 \]

### Step 8: Calculate the magnitude of the direction ratios
The magnitude of the direction ratios is:
\[ \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \]

### Step 9: Calculate the shortest distance
Now substituting into the distance formula:
\[ d = \frac{|1|}{\sqrt{3}} = \frac{1}{\sqrt{3}} \]

### Step 10: Final calculation for the shortest distance
The shortest distance from point \( P(1, 1, 1) \) to the line of intersection is:
\[ d = \sqrt{(1)^2 - \left(\frac{1}{\sqrt{3}}\right)^2} = \sqrt{1 - \frac{1}{3}} = \sqrt{\frac{2}{3}} = \frac{\sqrt{6}}{3} \]

Thus, the final answer for the shortest distance is:
\[ \text{Shortest distance} = \sqrt{\frac{8}{3}} \]