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

Options

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

Answer

To determine the type of conic section represented by the equation

\[ \frac{x^2}{1-k} - \frac{y^2}{1+k} = 1, \quad k > 1, \]

we will analyze the equation step by step.

### Step 1: Identify the terms in the equation
The equation is in the form of

\[ \frac{x^2}{A} - \frac{y^2}{B} = 1, \]

where \( A = 1-k \) and \( B = 1+k \).

### Step 2: Determine the signs of \( A \) and \( B \)
Since \( k > 1 \):
- \( 1 - k < 0 \) (because \( k \) is greater than 1, thus \( 1 - k \) is negative).
- \( 1 + k > 0 \) (since \( k \) is positive and greater than 1, \( 1 + k \) is positive).

### Step 3: Rewrite the equation
We can rewrite the equation as:

\[ -\frac{x^2}{k-1} + \frac{y^2}{1+k} = 1. \]

This can be rearranged to:

\[ \frac{y^2}{1+k} - \frac{x^2}{k-1} = 1. \]

### Step 4: Identify the conic section
The rewritten equation is of the form:

\[ \frac{y^2}{B} - \frac{x^2}{A} = 1, \]

where \( B = 1+k > 0 \) and \( A = k-1 > 0 \). This matches the standard form of a hyperbola:

\[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1. \]

### Conclusion
Since the equation matches the standard form of a hyperbola, we conclude that the given equation represents a hyperbola.

### Final Answer
The equation \(\frac{x^2}{1-k} - \frac{y^2}{1+k} = 1\) represents a hyperbola.

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