+206 Find the area between two concentric circles defined by
x2 + y2 -2x + 4y + 1 = 0
x2 + y2 -2x + 4y - 11 = 0
+128448 Let's put these into standard form, first
x^2 + y^2 -2x + 4y + 1 = 0
x^2 - 2x + y^2 + 2x = -1 complete the square on x and y
x^2 - 2x + 1 + y^2 + 2x + 4 = -1 + 1 + 4 factor
(x - 1)^2 + ( y + 2)^2 = 4
This is a circle centered at (1, -2) with a radius of 2
x^2 + y^2 -2x + 4y - 11 = 0
x^2 - 2x + y^2+ 4y = 11
x^2 - 2x + 1 + y^2 + 4y + 4 = 11 + 1 + 4
(x - 1)^2 + (y + 2)^2 = 16
This is a circle with the same center and a radius of 4
The area between the concentric circles =
pi [ 4^2 - 2^2] = pi [16 - 4 ] = 12pi units^2 ≈ 37.7 units^2
+26367 Find the area between two concentric circles defined by
Let xc the center of the circles in x
Let yc the center of the circles in y
\(x2 + y2 -2x + 4y \underbrace{+1}_{=x_c^2+y_c^2-r_1^2} = 0 \\\\ x2 + y2 -2x + 4y \underbrace{-11}_{=x_c^2+y_c^2-r_2^2} = 0 \)
\(\begin{array}{|lrcll|} \hline (1) & 1 &=& x_c^2+y_c^2-r_1^2 \\ (2) & -11 &=& x_c^2+y_c^2-r_2^2 \\ \hline (1)-(2): & 1-(-11) &=& x_c^2+y_c^2-r_1^2-(x_c^2+y_c^2-r_2^2) \\ & 1+11 &=& x_c^2+y_c^2-r_1^2-x_c^2-y_c^2+r_2^2 \\ & 12 &=& -r_1^2 +r_2^2 \\ & \mathbf{r_2^2-r_1^2} & \mathbf{=} & \mathbf{12} \\ \hline \end{array}\)
The area between two concentric circles:
\(\begin{array}{|rcll|} \hline A &=& \pi r_2^2 - \pi r_1^2 \\ &=& \pi \cdot ( r_2^2 - r_1^2) \quad & | \quad r_2^2-r_1^2 = 12 \\ &=& \pi \cdot 12 \\ &=& 37.6991118431 \\ \hline \end{array} \)
