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Consider the following statements about a common tangent to the parabola y^2 = 16\sqrt{3}\,x and the ellipse 2x^2 + y^2 = 4: Statement 1: One equation of such a common tangent is y = 2x + 2\sqrt{3}. Statement 2: If a line of the form y = mx + \frac{4\sqrt{3}}{m} \,(m \ne 0) is a common tangent to the parabola y^2 = 16\sqrt{3}\,x and the ellipse 2x^2 + y^2 = 4, then m^4 + 2m^2 = 24. Choose the correct option.

1. Statement 1 is false, and Statement 2 is true.
2. Both statements are true, and Statement 2 correctly explains Statement 1.
3. Both statements are true, but Statement 2 does not correctly explain Statement 1.
4. Statement 1 is true, and Statement 2 is false.

Correct answer: Statement 1 is true, and Statement 2 is false.

Solution

Statement 1 is correct because the given line is indeed a common tangent to both the parabola and the ellipse. However, Statement 2 is false as the derived equation does not hold true for the specified form of the tangent line.

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