The equation `(x^2)/(1-k)-(y^2)/(1+k)=1, k gt 1` represents:-

Options

A. circle
B. ellipse
C. hyperbola
D. None of these

Answer

To determine what the equation \(\frac{x^2}{1-k} - \frac{y^2}{1+k} = 1\) represents when \(k > 1\), we can analyze the equation step by step.

### Step 1: Analyze the coefficients
Given the equation:
\[ \frac{x^2}{1-k} - \frac{y^2}{1+k} = 1 \]
we note that \(k > 1\). This implies:
- \(1 - k < 0\) (since \(k\) is greater than 1, \(1-k\) is negative)
- \(1 + k > 0\) (since \(k\) is greater than 1, \(1+k\) is positive)

### Step 2: Rewrite the equation
We can rewrite the equation using the negative sign for the first term:
\[ -\frac{x^2}{k-1} - \frac{y^2}{1+k} = 1 \]
This can be rearranged to:
\[ -\frac{x^2}{k-1} = 1 + \frac{y^2}{1+k} \]
or equivalently:
\[ \frac{x^2}{k-1} - \frac{y^2}{1+k} = -1 \]

### Step 3: Identify the conic section
The standard form of a hyperbola is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
In our case, we have:
\[ \frac{x^2}{-(k-1)} - \frac{y^2}{1+k} = 1 \]
This indicates that the equation is of the form of a hyperbola, but with a negative sign on the \(x^2\) term.

### Step 4: Conclusion
Since the equation can be rearranged to fit the form of a hyperbola, we conclude that the equation \(\frac{x^2}{1-k} - \frac{y^2}{1+k} = 1\) represents a hyperbola when \(k > 1\).

### Final Answer
The equation represents a hyperbola.