If `x + y+z = 13, x^2 + y^2 + z^2 = 91` and `xz=y^2` then find the difference between x and z.

Options

A. 10
B. 9
C. 12
D. 8

Answer

To solve the problem step by step, we will use the given equations and relationships.

### Step 1: Write down the given equations.
We have the following equations:
1. \( x + y + z = 13 \) (Equation 1)
2. \( x^2 + y^2 + z^2 = 91 \) (Equation 2)
3. \( xz = y^2 \) (Equation 3)

### Step 2: Use the identity for the square of a sum.
We can use the identity:
\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \]
Substituting the values from Equations 1 and 2:
\[ 13^2 = 91 + 2(xy + yz + zx) \]
Calculating \( 13^2 \):
\[ 169 = 91 + 2(xy + yz + zx) \]
Rearranging gives:
\[ 169 - 91 = 2(xy + yz + zx) \]
\[ 78 = 2(xy + yz + zx) \]
Dividing by 2:
\[ xy + yz + zx = 39 \quad \text{(Equation 4)} \]

### Step 3: Substitute \( y^2 \) from Equation 3 into Equation 4.
From Equation 3, we have \( y^2 = xz \). We can express \( y \) in terms of \( x \) and \( z \):
\[ y = \sqrt{xz} \]
Substituting \( y \) into Equation 4:
\[ x\sqrt{xz} + \sqrt{xz}z + xz = 39 \]
Factoring out \( \sqrt{xz} \):
\[ \sqrt{xz}(x + z) + xz = 39 \]

### Step 4: Use Equation 1 to express \( x + z \).
From Equation 1, we can express \( x + z \):
\[ x + z = 13 - y \]
Substituting \( y = \sqrt{xz} \):
\[ x + z = 13 - \sqrt{xz} \]

### Step 5: Substitute \( x + z \) back into the equation.
Substituting \( x + z \) into the equation:
\[ \sqrt{xz}(13 - \sqrt{xz}) + xz = 39 \]
Let \( t = \sqrt{xz} \), then:
\[ t(13 - t) + t^2 = 39 \]
Expanding and rearranging gives:
\[ 13t - t^2 + t^2 = 39 \]
\[ 13t = 39 \]
Dividing by 13:
\[ t = 3 \]

### Step 6: Find \( xz \).
Since \( t = \sqrt{xz} = 3 \):
\[ xz = t^2 = 9 \quad \text{(Equation 5)} \]

### Step 7: Substitute \( xz \) back into Equation 1.
Now we have \( xz = 9 \) and can substitute it back into Equation 1:
\[ x + z = 13 - y \]
Using \( y = 3 \):
\[ x + z = 13 - 3 = 10 \quad \text{(Equation 6)} \]

### Step 8: Solve the system of equations.
Now we have two equations:
1. \( x + z = 10 \) (Equation 6)
2. \( xz = 9 \) (Equation 5)

Let \( z = 10 - x \). Substitute into Equation 5:
\[ x(10 - x) = 9 \]
Expanding gives:
\[ 10x - x^2 = 9 \]
Rearranging gives:
\[ x^2 - 10x + 9 = 0 \]

### Step 9: Factor the quadratic equation.
Factoring:
\[ (x - 9)(x - 1) = 0 \]
Thus, \( x = 9 \) or \( x = 1 \).

### Step 10: Find corresponding values of \( z \).
If \( x = 9 \):
\[ z = 10 - 9 = 1 \]
If \( x = 1 \):
\[ z = 10 - 1 = 9 \]

### Step 11: Find the difference between \( x \) and \( z \).
In both cases:
1. \( |9 - 1| = 8 \)
2. \( |1 - 9| = 8 \)

Thus, the difference between \( x \) and \( z \) is:
\[ \text{Difference} = 8 \]

### Final Answer:
The difference between \( x \) and \( z \) is \( 8 \).

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