For hyperbola x^2/9 - y^2/16 = 1, determine a^2, b^2, c^2 and coordinates of foci.

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Multiple Choice

For hyperbola x^2/9 - y^2/16 = 1, determine a^2, b^2, c^2 and coordinates of foci.

Explanation:
For a hyperbola in standard form x^2/a^2 − y^2/b^2 = 1, the denominators give a^2 and b^2 directly, the transverse axis is horizontal, and the foci are at (±c, 0) with c^2 = a^2 + b^2. Here the equation is x^2/9 − y^2/16 = 1, so a^2 = 9 and b^2 = 16. Then c^2 = a^2 + b^2 = 9 + 16 = 25, so c = 5. The foci lie on the x-axis at (±5, 0). So the correct set is a^2 = 9, b^2 = 16, c^2 = 25, c = 5, with foci at (±5, 0). The foci on the x-axis confirm a horizontal hyperbola; choosing (0, ±5) would correspond to a vertical hyperbola, which doesn’t match the given equation.

For a hyperbola in standard form x^2/a^2 − y^2/b^2 = 1, the denominators give a^2 and b^2 directly, the transverse axis is horizontal, and the foci are at (±c, 0) with c^2 = a^2 + b^2.

Here the equation is x^2/9 − y^2/16 = 1, so a^2 = 9 and b^2 = 16. Then c^2 = a^2 + b^2 = 9 + 16 = 25, so c = 5. The foci lie on the x-axis at (±5, 0).

So the correct set is a^2 = 9, b^2 = 16, c^2 = 25, c = 5, with foci at (±5, 0). The foci on the x-axis confirm a horizontal hyperbola; choosing (0, ±5) would correspond to a vertical hyperbola, which doesn’t match the given equation.

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