Let `f(x)=x^(2)+2px+2q^(2) and g(x)=-x^(2)-2qx+q^(2)` (where `q ne0`). If `x in R` and the minimum value of `f(x)` is equal to the maximum value of `g(x)`, then the value of `(p^(2))/(q^(2))` is equal to

Answer

To solve the problem, we need to find the value of \( \frac{p^2}{q^2} \) given the functions:

\[ f(x) = x^2 + 2px + 2q^2 \]
\[ g(x) = -x^2 - 2qx + q^2 \]

where \( q \neq 0 \). We know that the minimum value of \( f(x) \) is equal to the maximum value of \( g(x) \).

### Step 1: Find the minimum value of \( f(x) \)

The function \( f(x) \) is a quadratic function that opens upwards (since the coefficient of \( x^2 \) is positive). The minimum value of a quadratic function \( ax^2 + bx + c \) can be found using the formula:

\[ \text{Minimum value} = -\frac{D}{4a} \]

where \( D \) is the discriminant given by \( D = b^2 - 4ac \).

For \( f(x) \):
- \( a = 1 \)
- \( b = 2p \)
- \( c = 2q^2 \)

Calculating the discriminant \( D \):

\[ D = (2p)^2 - 4 \cdot 1 \cdot 2q^2 = 4p^2 - 8q^2 \]

Now, substituting into the minimum value formula:

\[ \text{Minimum value of } f(x) = -\frac{4p^2 - 8q^2}{4} = -p^2 + 2q^2 \]

### Step 2: Find the maximum value of \( g(x) \)

The function \( g(x) \) is a quadratic function that opens downwards (since the coefficient of \( x^2 \) is negative). The maximum value of a quadratic function \( ax^2 + bx + c \) can be found using the formula:

\[ \text{Maximum value} = -\frac{D}{4a} \]

For \( g(x) \):
- \( a = -1 \)
- \( b = -2q \)
- \( c = q^2 \)

Calculating the discriminant \( D \):

\[ D = (-2q)^2 - 4 \cdot (-1) \cdot q^2 = 4q^2 + 4q^2 = 8q^2 \]

Now, substituting into the maximum value formula:

\[ \text{Maximum value of } g(x) = -\frac{8q^2}{4 \cdot (-1)} = 2q^2 \]

### Step 3: Set the minimum value of \( f(x) \) equal to the maximum value of \( g(x) \)

From the problem statement, we have:

\[ -p^2 + 2q^2 = 2q^2 \]

### Step 4: Solve for \( p^2 \)

Rearranging the equation gives:

\[ -p^2 = 0 \]

Thus,

\[ p^2 = 0 \]

### Step 5: Find \( \frac{p^2}{q^2} \)

Since \( p^2 = 0 \), we can substitute into the expression:

\[ \frac{p^2}{q^2} = \frac{0}{q^2} = 0 \]

### Conclusion

The value of \( \frac{p^2}{q^2} \) is:

\[ \boxed{0} \]