Consider the following statements in respect of hyperbola \frac{x^{2}}{\cos^{2}\theta}-\frac{y^{2}}{\sin^{2}\theta}=1 :<br>1. The two foci are independent of \theta.<br>2. The eccentricity is \sec \theta.<br>3. The distance between the two foci is 2 units.<br>How many of the statements given above are correct?

Explanation

Here, a^2 = \cos^2\theta and b^2 = \sin^2\theta. The eccentricity is e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \tan^2\theta} = \sec\theta (Statement 2 is true). The foci coordinates are (\pm ae, 0) = (\pm \cos\theta \cdot \sec\theta, 0) = (\pm 1, 0), which are independent of \theta (Statement 1 is true). The distance between the foci is 2ae = 2(1) = 2 units (Statement 3 is true). Thus, all three statements are correct.

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