Question

The equation of a circle is given by x 2 + 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: 2 2 2 ( ) t x At y a B   − + − =     . The dimensions of t is given as [T–1].

• A. A = [L–1 T], B = [LT–1]
• B. A = [LT], B = [L–1T–1]
• C. A = [L–1T–1], B = [LT–1]
• D. A = [L–1T–1], B = [LT]

Answer 1

Answer: (B)

A = [LT], B = [L–1T–1]

Answer 2

The equation of a circle centered at $(h, k)$ is $(x-h)^2 + (y-k)^2 = a^2$. The modified equation is interpreted as $(x - At)^2 + (y - \frac{t}{B})^2 = a^2$. This implies the new center is $(At, \frac{t}{B})$. For dimensional consistency, terms representing coordinates must have dimensions of length $[L]$. Given $[t] = [T^{-1}]$, we derive the dimensions of $A$ and $B$: For $At$: $[A] \times [t] = [L] \implies [A] \times [T^{-1}] = [L] \implies [A] = [LT]$. For $\frac{t}{B}$: $\frac{[t]}{[B]} = [L] \implies \frac{[T^{-1}]}{[B]} = [L] \implies [B] = [L^{-1} T^{-1}]$.