A comet (assumed to be in an elliptical orbit around the sun) is at a distance of $0.4 \mathrm{AU}$ from the sun at the per ihelion. If the time period of the comet is 125 years, what is the aphelion distance ? AU : Astronomical Unit.

• A

  $50 \mathrm{AU}$
• B

  $25 \mathrm{AU}$
• Correct answer
  C

  $49.6 \mathrm{AU}$
• D

  $24.6 \mathrm{AU}$

Solution

$$ \therefore r=\frac{0.4+y}{2} $$ also $\left(\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}\right)^{2}=\left(\frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}\right)^{3}$ by Kepler's law of time- periods $\left(\mathrm{T}_{1}, \mathrm{r}_{1}\right.$ are taken for earth) $$ \therefore\left(\frac{1 \mathrm{y}}{125 \mathrm{y}}\right)^{2}=\left(\frac{1 \mathrm{AU}}{\left(\frac{0.4+\mathrm{y}}{2}\right) \mathrm{AU}}\right)^{3} $$ ving we get $\mathrm{y}=49.6 \mathrm{AU}$

Hence, the correct answer is option: 'C'.