JEE 2023 Mathematics PYQ — The line is the directrix of the ellipse with focus . If the tang… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
The line x=8 is the directrix of the ellipse E:a2x2+b2y2=1 with focus (2,0). If the tangent to E at point P in the first quadrant passes through (0,43) and intersects the x-axis at Q, then (3PQ)2 is equal to:
Choose the correct answer:
A.
39
(Correct Answer)
B.
40
C.
41
D.
42
Correct Answer:
39
Explanation
Solution:
Ellipse Properties: Focus ae=2, Directrix a/e=8.
Multiplying: a2=16⟹a=4.
e2=2/8=1/4⟹b2=a2(1−e2)=16(3/4)=12.
Ellipse: 16x2+12y2=1
Tangent: Tangent at (x1,y1) passes through (0,43).
16x⋅0+12y⋅43=1⟹y1=3.
Since P is on ellipse: 16x12+123=1⟹x12=12⟹x1=23.