If `H=(x^(2))/(a^(2))-(y^(2))/(b^(2))-1=0, C=(x^(2))/(a^(2))-(y^(2))/(b^(2))+1=0` and `A=(x^(2))/(a^(2))-(y^(2))/(b^(2))=0` then H, A and C are in

Options

A. (a) AP
B. (b) GP
C. (c) HP
D. (d) AGP

Answer

To determine the relationship between the expressions \( H \), \( A \), and \( C \), we start with the given equations:

1. \( H = \frac{x^2}{a^2} - \frac{y^2}{b^2} - 1 = 0 \)
2. \( C = \frac{x^2}{a^2} - \frac{y^2}{b^2} + 1 = 0 \)
3. \( A = \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \)

### Step 1: Rewrite the expressions
Let’s denote the common term \( \frac{x^2}{a^2} - \frac{y^2}{b^2} \) as \( \alpha \).

- From \( H \):
\[ H = \alpha - 1 = 0 \implies \alpha = 1 \]

- From \( A \):
\[ A = \alpha = 0 \implies \alpha = 0 \]

- From \( C \):
\[ C = \alpha + 1 = 0 \implies \alpha = -1 \]

### Step 2: Analyze the expressions
Now we can express \( H \), \( A \), and \( C \) in terms of \( \alpha \):

- \( H = \alpha - 1 \)
- \( A = \alpha \)
- \( C = \alpha + 1 \)

### Step 3: Check for Arithmetic Progression (AP)
For \( H \), \( A \), and \( C \) to be in arithmetic progression, the following condition must hold:
\[ 2A = H + C \]

Substituting the expressions:
\[ 2\alpha = (\alpha - 1) + (\alpha + 1) \]

### Step 4: Simplify the equation
Simplifying the right side:
\[ 2\alpha = \alpha - 1 + \alpha + 1 \]
\[ 2\alpha = 2\alpha \]

### Conclusion
Since both sides of the equation are equal, we conclude that \( H \), \( A \), and \( C \) are indeed in arithmetic progression.

Thus, the final answer is:
**\( H \), \( A \), and \( C \) are in Arithmetic Progression (AP).**

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