Given `y^(2)+z^(2)=ayz, z^(2)+x^(2)=bxz, x^(2)+y^(2)=cxy," express "(y^(2))/(xz) +(xz)/(y^(2))` in terms of a,b,c=….

Answer

To solve the problem, we need to express \( \frac{y^2}{xz} + \frac{xz}{y^2} \) in terms of \( a, b, c \) given the equations:

1. \( y^2 + z^2 = ayz \) (Equation 1)
2. \( z^2 + x^2 = bzx \) (Equation 2)
3. \( x^2 + y^2 = cxy \) (Equation 3)

### Step-by-Step Solution:

**Step 1: Rewrite the equations.**

From the given equations, we can express \( y^2 \), \( z^2 \), and \( x^2 \) in terms of \( a, b, c \):
- From Equation 1:
\[ y^2 = ayz - z^2 \]

- From Equation 2:
\[ z^2 = bzx - x^2 \]

- From Equation 3:
\[ x^2 = cxy - y^2 \]

**Step 2: Multiply the first equation by \( x^2 + y^2 \).**

We will multiply Equation 1 by \( x^2 + y^2 \):
\[ (y^2 + z^2)(x^2 + y^2) = ayz(x^2 + y^2) \]

Expanding both sides:
\[ y^2x^2 + y^4 + z^2x^2 + y^2z^2 = ayzx^2 + ay^2z \]

**Step 3: Divide by \( y^2xz \).**

Now, we need to find \( \frac{y^2}{xz} + \frac{xz}{y^2} \):
\[ \frac{y^2x^2 + y^4 + z^2x^2 + y^2z^2}{y^2xz} = \frac{ayzx^2 + ay^2z}{y^2xz} \]

This simplifies to:
\[ \frac{y^2}{xz} + \frac{xz}{y^2} + \frac{z^2}{xz} + \frac{y^2}{xz} = a \frac{x^2}{xz} + a \frac{y^2}{xz} \]

**Step 4: Combine terms.**

Now we can combine the terms:
\[ \frac{y^2}{xz} + \frac{xz}{y^2} = a \frac{x}{z} + a \frac{y}{z} \]

**Step 5: Substitute values from the equations.**

From the previous equations, we can substitute \( a, b, c \) into our expression:
\[ \frac{y^2}{xz} + \frac{xz}{y^2} = b + c - a \]

### Final Expression:
Thus, the expression \( \frac{y^2}{xz} + \frac{xz}{y^2} \) in terms of \( a, b, c \) is:
\[ \frac{y^2}{xz} + \frac{xz}{y^2} = b + c - a \]