The ratio of area of the concentric circles x^2+y^2-6x+12y+15=0 and x^2+y^2-6x+12y-15=0 is

Steps to Solve

1. Identify the equations of the circles

The given equations of the circles are: $$ x^2 + y^2 - 6x + 12y + 15 = 0 $$ $$ x^2 + y^2 - 6x + 12y - 15 = 0 $$

2. Rewrite the equations in standard form

To express the equations in standard form, complete the square for both the $x$ and $y$ terms.

For the first circle: $$ x^2 - 6x + y^2 + 12y + 15 = 0 $$ Completing the square on $x$: $$ (x - 3)^2 - 9 + (y + 6)^2 - 36 + 15 = 0 $$ Combine constants: $$ (x - 3)^2 + (y + 6)^2 = 30 $$

For the second circle: $$ x^2 - 6x + y^2 + 12y - 15 = 0 $$ Completing the square: $$ (x - 3)^2 - 9 + (y + 6)^2 - 36 - 15 = 0 $$ Combine constants: $$ (x - 3)^2 + (y + 6)^2 = 60 $$

3. Extract the radii of the circles

From the standard forms:

• The first circle has a radius $r_1 = \sqrt{30}$
• The second circle has a radius $r_2 = \sqrt{60}$

4. Calculate the areas of the circles

The area (A) of a circle is given by: $$ A = \pi r^2 $$

Calculating the area of the first circle: $$ A_1 = \pi (\sqrt{30})^2 = 30\pi $$

Calculating the area of the second circle: $$ A_2 = \pi (\sqrt{60})^2 = 60\pi $$

5. Find the ratio of the areas

The ratio of the areas of the circles is: $$ \text{Ratio} = \frac{A_1}{A_2} = \frac{30\pi}{60\pi} = \frac{30}{60} = \frac{1}{2} $$