If `x^2/a^2+y^2/b^2=1(a>b)` and `x^2-y^2=c^2` cut at right angles, then:

Options

A. `a^(2)+b^(2)=2c^(2)`
B. `b^(2)-a^(2)=2c^(2)`
C. `a^(2)-b^(2)=2c^(2)`
D. `a^(2)b^(2)=2c^(2)`

Answer

To solve the problem, we need to find the relationship between \(a\), \(b\), and \(c\) given that the equations of an ellipse and a hyperbola intersect at right angles.

### Step-by-Step Solution:

1. **Identify the Equations**:
The given equations are:
- Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (where \(a > b\))
- Hyperbola: \(x^2 - y^2 = c^2\)

2. **Understand the Geometry**:
Since the ellipse is centered at the origin and has a horizontal major axis, its foci are located at \((\pm c_e, 0)\) where \(c_e = \sqrt{a^2 - b^2}\).
The hyperbola also has foci at \((\pm c_h, 0)\) where \(c_h = a_h \sqrt{2}\) for a rectangular hyperbola, and here \(a_h = c\).

3. **Orthogonality Condition**:
The condition that these two curves cut each other at right angles implies that they are confocal. This means they share the same foci. Therefore, the foci of the ellipse and hyperbola must be equal:
\[ c_e = c_h \]

4. **Set the Foci Equal**:
From the ellipse:
\[ c_e = \sqrt{a^2 - b^2} \]
From the hyperbola:
\[ c_h = c\sqrt{2} \]
Setting these equal gives:
\[ \sqrt{a^2 - b^2} = c\sqrt{2} \]

5. **Square Both Sides**:
Squaring both sides to eliminate the square root:
\[ a^2 - b^2 = 2c^2 \]

6. **Final Result**:
This gives us the required relationship:
\[ a^2 - b^2 = 2c^2 \]

### Conclusion:
The relationship between \(a\), \(b\), and \(c\) is:
\[ \boxed{a^2 - b^2 = 2c^2} \]