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Ellipse question

2022 · Shift 1 · Q36
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  5. /2022 · Shift 1 · Q36

Ellipse question

2022 · Shift 1 · Q36

JEE AdvancedMathematicsEllipseMCQ+3 / −1

Consider the ellipse

x24+y23=1\frac{x^{2}}{4}+\frac{y^{2}}{3}=14x2​+3y2​=1

Let H(α,0),0<α<2H(\alpha, 0), 0\lt \alpha\lt 2H(α,0),0<α<2, be a point. A straight line drawn through HHH parallel to the yyy-axis crosses the ellipse and its auxiliary circle at points EEE and FFF respectively, in the first quadrant. The tangent to the ellipse at the point EEE intersects the positive xxx-axis at a point GGG. Suppose the straight line joining FFF and the origin makes an angle ϕ\phiϕ with the positive xxx-axis.

List-I List-II
(I) If ϕ=π4\phi=\frac{\pi}{4}ϕ=4π​, then the area of the triangle FGHF G HFGH is (P) (3−1)48\frac{(\sqrt{3}-1)^{4}}{8}8(3​−1)4​
(II) If ϕ=π3\phi=\frac{\pi}{3}ϕ=3π​, then the area of the triangle FGHF G HFGH is (Q) 1
(III) If ϕ=π6\phi=\frac{\pi}{6}ϕ=6π​, then the area of the triangle FGHF G HFGH is (R) 34\frac{3}{4}43​
(IV) If ϕ=π12\phi=\frac{\pi}{12}ϕ=12π​, then the area of the triangle FGHF G HFGH is (S) 123\frac{1}{2 \sqrt{3}}23​1​
(T) 332\frac{3 \sqrt{3}}{2}233​​

The correct option is:

  1. A
    (I)→(R);(II)→(S);(III)→(Q);(IV)→(P)(\mathrm{I}) \rightarrow(\mathrm{R}) ;(\mathrm{II}) \rightarrow(\mathrm{S}) ;(\mathrm{III}) \rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})(I)→(R);(II)→(S);(III)→(Q);(IV)→(P)
  2. B
    (I) →\rightarrow→(R); (II) →(T);(III)→(S);(IV)→(P)\rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})→(T);(III)→(S);(IV)→(P)
  3. C
    (I) →(Q);(II)→(T);(III)→(S);(IV)→(P)\rightarrow(\mathrm{Q}) ;(\mathrm{II}) \rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})→(Q);(II)→(T);(III)→(S);(IV)→(P)
  4. D
    (I) →\rightarrow→(Q); (II) →\rightarrow→(S); (III) →\rightarrow→(Q); (IV) →\rightarrow→ (P)
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