JEE 2023 Mathematics PYQ — If the tangent at a point on the parabola is parallel to the line… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
If the tangent at a point P on the parabola y2=3x is parallel to the line x+2y=1 and the tangents at the points Q and R on the ellipse 4x2+1y2=1 are perpendicular to the line x−y=2, then the area of the triangle PQR is:
Choose the correct answer:
A.
235
B.
35
(Correct Answer)
C.
59
D.
53
Correct Answer:
35
Explanation
Solution
1. Point P nikalne ke liye:
Parabola ki equation hai y2=3x. Iska slope nikalne ke liye differentiate karte hain:
2ydxdy=3⟹dxdy=2y3
Hume diya gaya hai ki tangent line x+2y=1 ke parallel hai. Is line ka slope m=−21 hai.
Isliye:
2y3=−21⟹y=−3
y=−3 ko y2=3x mein rakhne par:
(−3)2=3x⟹9=3x⟹x=3
Toh point P=(3,−3).
2. Points Q aur R nikalne ke liye:
Ellipse ki equation hai 4x2+1y2=1. Differentiate karne par:
42x+2ydxdy=0⟹dxdy=−4yx
Tangents line x−y=2 ke perpendicular hain. Is line ka slope 1 hai, toh perpendicular tangent ka slope m=−1 hoga.
−4yx=−1⟹x=4y
Isse ellipse ki equation mein rakhein:
4(4y)2+1y2=1⟹4y2+y2=1⟹5y2=1⟹y=±51
Agar y=51, toh x=54. Agar y=−51, toh x=−54.
Toh points hain Q=(54,51) aur R=(−54,−51).
3. Triangle PQR ka Area:
Area ka formula: Area=21∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣
