JEE 2023 Mathematics PYQ — Let the line meet the plane at point . If the angle between the l… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let the line ℓ:x=−21−y=λz−3,λ∈R meet the plane P:x+2y+3z=4 at point (α,β,γ). If the angle between the line ℓ and the plane P is cos−1(145), then α+2β+6γ is equal to _____ .
Choose the correct answer:
A.
11
(Correct Answer)
B.
12
C.
9
D.
8
Correct Answer:
11
Explanation
1. Line aur Plane ke beech ka Angle
Line ℓ ki equation ko standard form mein likhte hain:
1x−0=2y−1=λz−3
Line ke direction ratios (b) hain: (1,2,λ).
Plane P:x+2y+3z=4 ke normal ke direction ratios (n) hain: (1,2,3).
Line aur Plane ke beech ka angle θ hai, jahan sinθ=∣b∣∣n∣∣b⋅n∣.
Humein diya gaya hai θ=cos−1(145).
Iska matlab cos2θ=145.
Hum jaante hain sin2θ=1−cos2θ=1−145=149.
Isliye, sinθ=143.
Ab formula mein values rakhte hain:
143=12+22+λ212+22+32∣(1)(1)+(2)(2)+(λ)(3)∣
143=5+λ214∣5+3λ∣
35+λ2=∣5+3λ∣
Dono side square karne par:
9(5+λ2)=(5+3λ)2
45+9λ2=25+9λ2+30λ
20=30λ⟹λ=32.
2. Intersection Point (α,β,γ) nikalna
Line par koi bhi general point maante hain:
x=r,y=2r+1,z=32r+3.
Ye point plane x+2y+3z=4 par lie karta hai:
r+2(2r+1)+3(32r+3)=4
r+4r+2+2r+9=4
7r+11=4⟹7r=−7⟹r=−1.
Point (α,β,γ) ke coordinates honge:
α=−1
β=2(−1)+1=−1
γ=32(−1)+3=37.
3. Final Value nikalna
Humein α+2β+6γ ki value nikalni hai:
=(−1)+2(−1)+6(37)
=−1−2+14
=11.
Final Answer:
α+2β+6γ ki value 11 hai.
Explanation
1. Line aur Plane ke beech ka Angle
Line ℓ ki equation ko standard form mein likhte hain:
1x−0=2y−1=λz−3
Line ke direction ratios (b) hain: (1,2,λ).
Plane P:x+2y+3z=4 ke normal ke direction ratios (n) hain: (1,2,3).
Line aur Plane ke beech ka angle θ hai, jahan sinθ=∣b∣∣n∣∣b⋅n∣.
