EllipseMedium
Question
A rectangle PQRS is formed, where P,Q lie on y2 = 4x (0 < x ≤ 4) and R,S lie on y2 = - 4x (-4 ≤ x < 0). An ellipse 
= 1 (0 < a < b) passes through all the vertices of rectangle. If eccentricity of ellipse is 
, then -
= 1 (0 < a < b) passes through all the vertices of rectangle. If eccentricity of ellipse is , then - Options
A.a ∈ (0, 
]
]B.b ∈ (0, ]
]C.maximum area of ellipse is 24 √2 π sq. units
D.maximum area of rectangle PQRS is 16 sq. units.
Solution
Let any point on the parabola y2 = 4x be (2t, t2)
t ∈ [- 2, 2] - {0}
it also lies on
= 1, e2 = 1 - 
⇒ b2 = 2a2,
= 1⇒ a2 = t4 + 2t2, t ∈ [-2,2] - {0}
⇒ a2 ∈ (0, 24] ⇒ b2∈ (0, 48]
⇒ maximum area of ellipse = πab
= π.
. 
= 24√2πNow rectangle of maximum area is square whose area = 64 sq. units
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