If the cuves, `(x^(2))/(a)+(y^(2))/(b)=1 and (x^(2))/(c )+(y^(2))/(d)=1` intersect each other at an angle of `90^(@)`, then which of the following relations is TRUE?

Options

A. `a+ b=c+d`
B. `a-b=c-d`
C. `a-c=b+d`
D. `ab=(c+d)/(a+b)`

Answer

To solve the problem, we need to analyze the given equations of the curves and their intersection properties. The curves are given by:

1. \(\frac{x^2}{a} + \frac{y^2}{b} = 1\) (which represents an ellipse)
2. \(\frac{x^2}{c} + \frac{y^2}{d} = 1\) (which represents another conic)

Since the curves intersect at an angle of \(90^\circ\), we can use the property of orthogonal intersection of conics. The condition for two conics to intersect orthogonally is given by the relation involving their coefficients.

### Step-by-Step Solution:

1. **Identify the type of conics**:
- The first equation represents an ellipse.
- The second equation can also represent an ellipse or a hyperbola depending on the signs. For orthogonality, we will consider the second equation as a hyperbola.

2. **Write the equations in standard form**:
- The first equation is already in standard form for an ellipse.
- The second equation can be rewritten as:
\[ \frac{x^2}{c} - \frac{y^2}{d} = 1 \]
This is the standard form of a hyperbola.

3. **Orthogonality condition**:
- For two conics to intersect orthogonally, the following condition must hold:
\[ \frac{1}{a} + \frac{1}{c} = \frac{b}{d} \]
This is derived from the properties of the gradients of the tangents at the points of intersection.

4. **Substituting the values**:
- From the orthogonality condition, we can rearrange the terms to find the relationship between \(a\), \(b\), \(c\), and \(d\):
\[ \frac{1}{a} + \frac{1}{c} = \frac{b}{d} \]

5. **Final relation**:
- Rearranging gives us:
\[ d(a + c) = bc \]
This is the required relation that holds true for the curves to intersect orthogonally.

### Conclusion:
The relation that holds true for the curves to intersect at an angle of \(90^\circ\) is:
\[ d(a + c) = bc \]