JEE 2023 Mathematics PYQ — If the point lies on the curve traced by the mid-points of the li… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
If the point (α,373) lies on the curve traced by the mid-points of the line segments of the lines xcosθ+ysinθ=7,θ∈(0,2π) between the co-ordinates axes, then α is equal to
Choose the correct answer:
A.
73
B.
−7
C.
7
(Correct Answer)
D.
−73
Correct Answer:
7
Explanation
Step 1: Line ke Intercepts nikalna
Line ki equation di gayi hai:
xcosθ+ysinθ=7
Isse intercept form ax+by=1 mein convert karte hain:
7/cosθx+7/sinθy=1
Iska matlab line axes ko do points par milti hai:
X-axis par point A=(cosθ7,0)
Y-axis par point B=(0,sinθ7)
Step 2: Mid-point ka Locus nikalna
Maana mid-point M=(h,k) hai. Mid-point formula se:
h=2cosθ7+0=2cosθ7⟹cosθ=2h7
k=20+sinθ7=2sinθ7⟹sinθ=2k7
Hame pata hai ki sin2θ+cos2θ=1 hota hai. In values ko rakhte hain:
(2h7)2+(2k7)2=1
4h249+4k249=1⟹h21+k21=494
Toh curve ki general equation (x,y mein) hogi:
x21+y21=494
Step 3: Point (α,373) ko curve mein rakhna
Chunki yeh point curve par lie karta hai:
α21+(373)21=494
α21+949×31=494
α21+1479=494
α21+493=494
α21=494−493=491
Iska matlab:
α2=49⟹α=±7
Sawal mein θ∈(0,π/2) diya gaya hai (first quadrant), isliye coordinates positive honge. Option ke hisaab se:
α=7
Explanation
Step 1: Line ke Intercepts nikalna
Line ki equation di gayi hai:
xcosθ+ysinθ=7
Isse intercept form ax+by=1 mein convert karte hain:
7/cosθx+7/sinθy=1
Iska matlab line axes ko do points par milti hai:
X-axis par point A=(cosθ7,0)
Y-axis par point B=(0,sinθ7)
Step 2: Mid-point ka Locus nikalna
Maana mid-point M=(h,k) hai. Mid-point formula se:
h=2cosθ7+0=2cosθ7⟹cosθ=2h7
k=20+sinθ7=2sinθ7⟹sinθ=2k7
Hame pata hai ki sin2θ+cos2θ=1 hota hai. In values ko rakhte hain:
(2h7)2+(2k7)2=1
4h249+4k249=1⟹h21+k21=494
Toh curve ki general equation (x,y mein) hogi:
x21+y21=494
Step 3: Point (α,373) ko curve mein rakhna
Chunki yeh point curve par lie karta hai:
α21+(373)21=494
α21+949×31=494
α21+1479=494
α21+493=494
α21=494−493=491
Iska matlab:
α2=49⟹α=±7
Sawal mein θ∈(0,π/2) diya gaya hai (first quadrant), isliye coordinates positive honge. Option ke hisaab se:
