JEE 2023 Mathematics PYQ — A light ray emits from the origin making an angle with the positi… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
A light ray emits from the origin making an angle 30∘ with the positive x-axis. After getting reflected by the line x+y=1, if this ray intersects x-axis at Q, then the abscissa of Q is:
Choose the correct answer:
A.
2(3+1)3
B.
3+32
(Correct Answer)
C.
(3−1)2
D.
3−32
Correct Answer:
3+32
Explanation
Solving:
1. Incident Ray Equation:
Slope m=tan30∘=31. Origin (0,0) se guzarne wali line:
y=31x⟹x−3y=0
2. Point of Reflection (P):
Line x+y=1 aur x−3y=0 ka intersection:
(3y)+y=1⟹y=3+11
x=3+13⟹P=(3+13,3+11)
3. Reflected Ray Equation:
Line x+y=1 ka slope −1 hai (135∘).
Angle of incidence = 135∘−30∘=105∘.
Angle of reflection = 135∘+105∘=240∘ (ya simply 60∘ with negative x-axis).
Reflected ray ka slope: m′=tan(135∘+(135∘−30∘))=tan240∘=3.
4. Finding Q (x-intercept):
Equation: y−3+11=3(x−3+13)
For x-axis (Q), put y=0:
3+1−1=3x−3+13
3x=3+13−1=3+12
x=3(3+1)2=3+32
Correct Option: (2)
Explanation
Solving:
1. Incident Ray Equation:
Slope m=tan30∘=31. Origin (0,0) se guzarne wali line:
y=31x⟹x−3y=0
2. Point of Reflection (P):
Line x+y=1 aur x−3y=0 ka intersection:
(3y)+y=1⟹y=3+11
x=3+13⟹P=(3+13,3+11)
3. Reflected Ray Equation:
Line x+y=1 ka slope −1 hai (135∘).
Angle of incidence = 135∘−30∘=105∘.
Angle of reflection = 135∘+105∘=240∘ (ya simply 60∘ with negative x-axis).
Reflected ray ka slope: m′=tan(135∘+(135∘−30∘))=tan240∘=3.
