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Units and Measurements question

2023 · 29 Jan · Shift 2 · Q47
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Units and Measurements question

2023 · 29 Jan · Shift 2 · Q47

JEE MainPhysicsUnits and MeasurementsMCQ+4 / −1
The equation of a circle is given by x2+y2=a2x^2+y^2=a^2x2+y2=a2, where a is the radius. If the equation is modified to change the origin other than (0, 0), then find out the correct dimensions of A and B in a new equation : (x−At)2+(y−tB)2=a2{(x - At)^2} + {\left( {y - {t \over B}} \right)^2} = {a^2}(x−At)2+(y−Bt​)2=a2. The dimensions of t is given as [T−1][\mathrm{T^{-1}]}[T−1].
  1. A
    A=[L−1T−1],B=[LT−1]\mathrm{A=[L^{-1}T^{-1}],B=[LT^{-1}]}A=[L−1T−1],B=[LT−1]
  2. B
    A=[L−1T−1],B=[LT]\mathrm{A=[L^{-1}T^{-1}],B=[LT]}A=[L−1T−1],B=[LT]
  3. C
    A=[LT],B=[L−1T−1]\mathrm{A=[LT],B=[L^{-1}T^{-1}]}A=[LT],B=[L−1T−1]
  4. D
    A=[L−1T],B=[LT−1]\mathrm{A=[L^{-1}T],B=[LT^{-1}]}A=[L−1T],B=[LT−1]
View written solutionFree

Correct answer: C

  1. Use dimensional consistency of the circle equation

    Original equation: x2+y2=a2x^2+y^2=a^2x2+y2=a2

    Since aaa is the radius, its dimension is length: [a]=[L][a]=[L][a]=[L] Hence, [x]=[y]=[L][x]=[y]=[L][x]=[y]=[L]

  2. Analyze the modified equation

    (x−At)2+(y−tB)2=a2(x-At)^2 + \left(y-\frac{t}{B}\right)^2 = a^2(x−At)2+(y−Bt​)2=a2

    For subtraction to be valid, the quantities inside each bracket must have the same dimensions.

  3. Find dimension of AAA

    Since xxx has dimension [L][L][L], we must have: [At]=[L][At]=[L][At]=[L]

    Given: [t]=[T−1][t]=[T^{-1}][t]=[T−1]

    Therefore, [A]=[L][T−1]=[L][T]=[LT][A]=\frac{[L]}{[T^{-1}]}=[L][T]=[LT][A]=[T−1][L]​=[L][T]=[LT]

  4. Find dimension of BBB

    Since yyy has dimension [L][L][L], we need: [tB]=[L]\left[\frac{t}{B}\right]=[L][Bt​]=[L]

    So, [B]=[t][L]=[T−1][L]=[L−1T−1][B]=\frac{[t]}{[L]}=\frac{[T^{-1}]}{[L]}=[L^{-1}T^{-1}][B]=[L][t]​=[L][T−1]​=[L−1T−1]

  5. Match with options

    Thus, A=[LT],B=[L−1T−1]A=[LT], \qquad B=[L^{-1}T^{-1}]A=[LT],B=[L−1T−1]

    This matches Option C.

  6. Comparison with stored answer

    Stored correct answer: C

    Our derived answer is also C, so they agree.

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