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Question:

Statement 1: An equation of a common tangent to the parabola y² = 16√3x and the ellipse 2x² + y² = 4 is y = 2x + 2√3. Statement 2: If the line y = mx + 4√3m, (m ≠ 0) is a common tangent to the parabola y² = 16√3x and the ellipse 2x² + y² = 4, then m satisfies m⁴ + 2m² = 24. Which of the following is correct?

Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

Statement 1 is false, Statement 2 is true.

Statement 1 is true , Statement 2 is false.

Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1

Solution:

Equation of the parabola y² = 16√3x
Equation of the ellipse 2x² + y² = 4
and common tangent is y = 2x + 2√3
Hence, the common tangent is y = mx + 4√3m
Condition of tangency in ellipse is c² = a²m² + b²
48m² = 2m² + 4 ⇒ m⁴ + 2m² = 24