The lines `x=y=z` meets the plane `x+y+z=1` at the point P and the sphere `x^2+y^2+z^2=1` at the point R and S, then

Options

A. `PR+PS=2`
B. `PRtimesPS=(2)/(3)`
C. `PR=PS`
D. `PR+PS=RS`

Answer

To solve the problem, we need to find the points where the line \( x = y = z \) intersects the plane \( x + y + z = 1 \) and the sphere \( x^2 + y^2 + z^2 = 1 \).

### Step 1: Find the intersection point P with the plane
The line is given by the equations:
\[ x = y = z \]

Substituting \( x \) for \( y \) and \( z \) in the plane equation:
\[ x + y + z = 1 \]
becomes:
\[ x + x + x = 1 \]
or:
\[ 3x = 1 \]
Thus:
\[ x = \frac{1}{3} \]

Since \( x = y = z \), we have:
\[ y = \frac{1}{3}, \quad z = \frac{1}{3} \]

So, the coordinates of point \( P \) are:
\[ P\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \]

### Step 2: Find the intersection points R and S with the sphere
The sphere is defined by the equation:
\[ x^2 + y^2 + z^2 = 1 \]

Again substituting \( x = y = z \):
\[ x^2 + x^2 + x^2 = 1 \]
This simplifies to:
\[ 3x^2 = 1 \]
Thus:
\[ x^2 = \frac{1}{3} \]
Taking the square root gives:
\[ x = \pm \frac{1}{\sqrt{3}} \]

Since \( x = y = z \), we have two points:
1. For \( x = \frac{1}{\sqrt{3}} \):
\[ R\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \]
2. For \( x = -\frac{1}{\sqrt{3}} \):
\[ S\left(-\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}\right) \]

### Step 3: Calculate the distances PR, PS, and RS
1. **Distance \( PR \)**:
\[ PR = \sqrt{\left(\frac{1}{\sqrt{3}} - \frac{1}{3}\right)^2 + \left(\frac{1}{\sqrt{3}} - \frac{1}{3}\right)^2 + \left(\frac{1}{\sqrt{3}} - \frac{1}{3}\right)^2} \]
Simplifying:
\[ = \sqrt{3\left(\frac{1}{\sqrt{3}} - \frac{1}{3}\right)^2} \]
\[ = \sqrt{3} \left(\frac{1}{\sqrt{3}} - \frac{1}{3}\right) \]

2. **Distance \( PS \)**:
\[ PS = \sqrt{\left(-\frac{1}{\sqrt{3}} - \frac{1}{3}\right)^2 + \left(-\frac{1}{\sqrt{3}} - \frac{1}{3}\right)^2 + \left(-\frac{1}{\sqrt{3}} - \frac{1}{3}\right)^2} \]
Similar simplification as above.

3. **Distance \( RS \)**:
\[ RS = \sqrt{\left(\frac{1}{\sqrt{3}} - \left(-\frac{1}{\sqrt{3}}\right)\right)^2 + \left(\frac{1}{\sqrt{3}} - \left(-\frac{1}{\sqrt{3}}\right)\right)^2 + \left(\frac{1}{\sqrt{3}} - \left(-\frac{1}{\sqrt{3}}\right)\right)^2} \]
This simplifies to:
\[ = 2\sqrt{3} \]

### Final Results
- Point \( P \) is \( \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) \)
- Point \( R \) is \( \left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \)
- Point \( S \) is \( \left(-\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}\right) \)