To the circle x 2 + y 2 = 4 two tangents are drawn from P ( − 4 , 0 ) which touch the circle at T 1 and T 2 and a rhombus P T 1 P ′ T 2 is completed. If P is taken to be at ( h , 0 ) such that

Question

To the circle x 2 + y 2 = 4 two tangents are drawn from P ( − 4 , 0 ) which touch the circle at T 1 ​ and T 2 ​ and a rhombus P T 1 ​ P ′ T 2 ​ is completed. If P is taken to be at ( h , 0 ) such that P ′ lies on the circle, the area of the rhombus is

TestHub · Accepted Answer

For P ( h , 0 ) the chord of contact is x = h 4 ​ , meeting the x-axis at M ( h 4 ​ , 0 ) , so P ′ = ( h 8 ​ − h , 0 ) . P ′ lies on the circle: h 8 ​ − h = ± 2 ⇒ h = ± 4 (as ∣ h ∣ > 2 ). Take h = 4 : chord x = 1 , T 1 ​ T 2 ​ = 2 3 ​ , P P ′ = 6 . Area of rhombus = 2 1 ​ × P P ′ × T 1 ​ T 2 ​ = 2 1 ​ × 6 × 2 3 ​ = 6 3 ​ .

Mathematics - Circle Question with Solution | TestHub

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