If x,y,z `in` R , x +y+z =5
`x^(2) +y^(2)+z^(2)=9` then length of interval in which x lies is

Options

A. `8//3`
B. `4//3`
C. `2//3`
D. `1//3`

Answer

To solve the problem step by step, we start with the given equations:

1. \( x + y + z = 5 \)
2. \( x^2 + y^2 + z^2 = 9 \)

### Step 1: Express \( y + z \) and \( y^2 + z^2 \) in terms of \( x \)

From the first equation, we can express \( y + z \) as:
\[ y + z = 5 - x \]

From the second equation, we can express \( y^2 + z^2 \) as:
\[ y^2 + z^2 = 9 - x^2 \]

### Step 2: Use the identity for \( (y + z)^2 \)

We know that:
\[ (y + z)^2 = y^2 + z^2 + 2yz \]
Substituting the expressions we found:
\[ (5 - x)^2 = (9 - x^2) + 2yz \]

### Step 3: Expand and rearrange the equation

Expanding \( (5 - x)^2 \):
\[ 25 - 10x + x^2 = 9 - x^2 + 2yz \]
Rearranging gives:
\[ 2yz = 25 - 10x + x^2 - 9 + x^2 \]
\[ 2yz = 2x^2 - 10x + 16 \]
\[ yz = x^2 - 5x + 8 \]

### Step 4: Set up a quadratic equation

Now we can express \( y \) and \( z \) as roots of the quadratic equation:
\[ t^2 - (5 - x)t + (x^2 - 5x + 8) = 0 \]
where \( t \) represents either \( y \) or \( z \).

### Step 5: Find the discriminant

For \( y \) and \( z \) to be real numbers, the discriminant of this quadratic must be non-negative:
\[ D = (5 - x)^2 - 4(x^2 - 5x + 8) \geq 0 \]

Expanding the discriminant:
\[ D = (25 - 10x + x^2) - 4(x^2 - 5x + 8) \]
\[ D = 25 - 10x + x^2 - 4x^2 + 20x - 32 \]
\[ D = -3x^2 + 10x - 7 \geq 0 \]

### Step 6: Solve the quadratic inequality

To find the values of \( x \) for which \( D \geq 0 \), we first find the roots of the equation:
\[ -3x^2 + 10x - 7 = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-10 \pm \sqrt{10^2 - 4(-3)(-7)}}{2(-3)} \]
\[ x = \frac{-10 \pm \sqrt{100 - 84}}{-6} = \frac{-10 \pm \sqrt{16}}{-6} \]
\[ x = \frac{-10 \pm 4}{-6} \]
Calculating the roots:
\[ x_1 = \frac{-6}{-6} = 1, \quad x_2 = \frac{-14}{-6} = \frac{7}{3} \]

### Step 7: Determine the interval

The quadratic opens downwards (since the coefficient of \( x^2 \) is negative), so the inequality \( -3x^2 + 10x - 7 \geq 0 \) holds between the roots:
\[ 1 \leq x \leq \frac{7}{3} \]

### Step 8: Calculate the length of the interval

The length of the interval where \( x \) lies is:
\[ \frac{7}{3} - 1 = \frac{7}{3} - \frac{3}{3} = \frac{4}{3} \]

Thus, the final answer is:
\[ \text{Length of interval in which } x \text{ lies is } \frac{4}{3}. \]