If the area enclosed by y 2 = 2 x and x 2 + 4 + 4 x = 4 y 2 is k square units, then the value of 3 k is equal to

Question

If the area enclosed by y 2 = 2 x and x 2 + 4 + 4 x = 4 y 2 is k square units, then the value of 3 k is equal to,

MARKS App · Accepted Answer

Given equation is x + 2 2 = 4 y 2 or x + 2 = 2 y and x + 2 = - 2 y Solving x + 2 = 2 y with the equation of the parabola, we get, y 2 = 2 2 y - 2 ⇒ y 2 - 4 y + 4 = 0 ⇒ y = 2 Hence, the pair of lines are tangent to the parabola. So, k = 2 ∫ 0 2 y 2 2 d y - ∫ 0 2 2 y - 2 d y = 2 y 3 6 - y 2 + 2 y 0 2 = 2 8 6 = 8 3 ∴ 3 k = 8

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