Question

If the radius of a circle becomes \( k \) times, then the ratio of the areas of the previous and new circles is:

• A. \( 1 : k \)
• B. \( 2 : k^3 \)
• C. \( 1 : k^2 \)
• D. \( k^2 : 1 \)

Hint:

The area of a circle is proportional to the square of its radius. So, if the radius increases by a factor of \( k \), the area increases by a factor of \( k^2 \).

Answer 1

The Correct Option is C

The area \( A \) of a circle is given by the formula: \[ A = \pi r^2, \] where \( r \) is the radius. If the radius increases by a factor of \( k \), then the new area becomes: \[ A_{\text{new}} = \pi (kr)^2 = k^2 \pi r^2. \] Therefore, the ratio of the areas of the new and previous circles is: \[ \frac{A_{\text{new}}}{A_{\text{old}}} = \frac{k^2 \pi r^2}{\pi r^2} = k^2. \] Thus, the ratio of the areas of the previous and new circles is \( \boxed{1 : k^2} \).