JEE 2022 Mathematics PYQ — Let be a focal chord of the parabola such that it subtends an ang… | Mathem Solvex | Mathem Solvex
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JEE 2022 — Mathematics PYQ
JEE | Mathematics | 2022
Let PQ be a focal chord of the parabola y2=4x such that it subtends an angle of 2π at the point (3,0). Let the line segment PQ be also a focal chord of the ellipse E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a^2 > b^2. If e is the eccentricity of the ellipse E, then the value of e21 is equal to :
Choose the correct answer:
A.
1+2
B.
3+22
(Correct Answer)
C.
1+23
D.
4+53
Correct Answer:
3+22
Explanation
Solution
Parabola Analysis: For y2=4x, the focus is (1,0). Let P=(t2,2t) and Q=(1/t2,−2/t).
Angle Condition: The slopes from P and Q to (3,0) must satisfy mP⋅mQ=−1.
Focal Chord Length: For t2=1, P=(1,2) and Q=(1,−2). The length of the focal chord PQ is 4.
Ellipse Analysis: A focal chord of an ellipse passing through focus (ae,0) has length a2sin2θ+b2cos2θ2ab2. Here the chord PQ is vertical (x=1), so θ=π/2 and its length is a2b2=4.
Also, since it is a focal chord, the focus must be at x=1, so ae=1.
Relation:b2=a2(1−e2). Substituting into length formula: a2a2(1−e2)=4⟹a(1−e2)=2.
Since a=1/e, we have e1(1−e2)=2⟹1−e2=2e⟹e2+2e−1=0.
Solving for e: e=2−2±4+4=−1±2. Since e > 0, e=2−1.
Final Value:e21=(2−1)21=2+1−221=3−221.
Rationalizing: 9−83+22=3+22.
Final Answer: (B)
Explanation
Solution
Parabola Analysis: For y2=4x, the focus is (1,0). Let P=(t2,2t) and Q=(1/t2,−2/t).
Angle Condition: The slopes from P and Q to (3,0) must satisfy mP⋅mQ=−1.
Focal Chord Length: For t2=1, P=(1,2) and Q=(1,−2). The length of the focal chord PQ is 4.
Ellipse Analysis: A focal chord of an ellipse passing through focus (ae,0) has length a2sin2θ+b2cos2θ2ab2. Here the chord PQ is vertical (x=1), so θ=π/2 and its length is a2b2=4.
Also, since it is a focal chord, the focus must be at x=1, so ae=1.
Relation:b2=a2(1−e2). Substituting into length formula: a2a2(1−e2)=4⟹a(1−e2)=2.
Since a=1/e, we have e1(1−e2)=2⟹1−e2=2e⟹e2+2e−1=0.
Solving for e: e=2−2±4+4=−1±2. Since e > 0, e=2−1.
