Consider the following in respect of the equation \frac{x^{2}}{24-k}+\frac{y^{2}}{k-16}=2.<br>1. The equation represents an ellipse if k=19.<br>2. The equation represents a hyperbola if k=12.<br>3. The equation represents a circle if k=20.<br>How many of the statements given above are correct?

Explanation

Check statement 1: For k=19, \frac{x^2}{5} + \frac{y^2}{3} = 2, which is an ellipse (both denominators positive). Check statement 2: For k=12, \frac{x^2}{12} + \frac{y^2}{-4} = 2, which is a hyperbola (one positive, one negative denominator). Check statement 3: For k=20, \frac{x^2}{4} + \frac{y^2}{4} = 2 \implies x^2+y^2=8, which represents a circle (equal positive denominators). Therefore, all three statements are correct.

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