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Question:

Given: A circle, 2x² + 2y² = 5 and a parabola, y² = 4√5x. Statement-I: An equation of a common tangent to these curves is y = x + √5. Statement-II: If the line, y = mx + √5/m (m ≠ 0) is their common tangent, then m satisfies m⁴ - m² + 2 = 0. Select the correct option:

Statement-I is true; Statement-II is true;Statement-II is not the correct explanation of Statement-I.

Statement-I is true; Statement-II is false.

Statement-I is false; Statement-II is true

Statement-I is true; Statement-II is true;Statement-II is the correct explanation of Statement-I.

Solution:

Let the tangent to the parabola be y = mx + √5/m (m ≠ 0). Now, its distance from the centre of the circle must be equal to the radius of the circle. So, |√5/m| = √5/√2 √(1 + m²) = (1 + m²) m² = 2 ⇒ m⁴ + m² -2 = 0. ⇒ (m² - 1)(m² + 2) = 0 ⇒ m = ±1 So, the common tangents are y = x + √5 and y = -x - √5.