Identify the graph of each of the following equations. \( \begin{array}{ll}\text { (a) } 4 x^{2}-8 x-49 y^{2}+196 y-388=0 & \text { (e) } 36 x^{2}+360 x+64 y^{2}-512 y+1924 \\ \text { (b) } x^{2}+5 x+y^{2}-y+7=0 & 0 \\ \text { (c) } y^{2}-48 x+6 y=-729 & \text { (f) } x^{2}+y^{2}-18 y-19=0 \\ \text { (d) } 49 x^{2}+196 x+100 y^{2}+1400 y+ & \text { (g) }-5 x^{2}+60 x+7 y^{2}+84 y+72= \\ 196=0 & \text { (h) } x^{2}-16 x+20 y=136\end{array} \)

To identify the graphs of the given equations, we will analyze each equation and determine its type (e.g., circle, ellipse, hyperbola, or parabola) by rewriting them in standard forms. ### (a) \( 4x^2 - 8x - 49y^2 + 196y - 388 = 0 \) 1. Rearranging gives: \[ 4x^2 - 8x - 49y^2 + 196y = 388 \] 2. Completing the square for \(x\) and \(y\): - For \(x\): \(4(x^2 - 2x) = 4((x-1)^2 - 1) = 4(x-1)^2 - 4\) - For \(y\): \(-49(y^2 - 4y) = -49((y-2)^2 - 4) = -49(y-2)^2 + 196\) 3. Substitute back: \[ 4(x-1)^2 - 4 - 49(y-2)^2 + 196 = 388 \] \[ 4(x-1)^2 - 49(y-2)^2 + 192 = 388 \] \[ 4(x-1)^2 - 49(y-2)^2 = 196 \] 4. Dividing by 196: \[ \frac{(x-1)^2}{49} - \frac{(y-2)^2}{4} = 1 \] This is a hyperbola. ### (b) \( x^2 + 5x + y^2 - y + 7 = 0 \) 1. Rearranging gives: \[ x^2 + 5x + y^2 - y = -7 \] 2. Completing the square: - For \(x\): \(x^2 + 5x = (x + \frac{5}{2})^2 - \frac{25}{4}\) - For \(y\): \(y^2 - y = (y - \frac{1}{2})^2 - \frac{1}{4}\) 3. Substitute back: \[ (x + \frac{5}{2})^2 - \frac{25}{4} + (y - \frac{1}{2})^2 - \frac{1}{4} = -7 \] \[ (x + \frac{5}{2})^2 + (y - \frac{1}{2})^2 = -7 + \frac{25}{4} + \frac{1}{4} = -7 + 6.5 = -0.5 \] This represents no real solutions (no graph). ### (c) \( y^2 - 48x + 6y = -729 \) 1. Rearranging gives: \[ y^2 + 6y = 48x - 729 \] 2. Completing the square for \(y\): \[ (y + 3)^2 - 9 = 48x - 729 \] \[ (y + 3)^2 = 48x - 720 \] \[ (y + 3)^2 = 48(x - 15) \] This is a parabola. ### (d) \( 49x^2 + 196x + 100y^2 + 1400y + 196 = 0 \) 1. Rearranging gives: \[ 49x^2 + 196x + 100y^2 + 1400y = -196 \] 2. Completing the square: - For \(x\): \(49(x^2 + 4x) = 49((x + 2)^2 - 4) = 49(x + 2)^2 - 196\) - For \(y\): \(100(y^2 + 14y) = 100((y + 7)^2 - 49) = 100(y + 7)^2 - 4900\) 3. Substitute back: \[ 49(x + 2)^2 - 196 + 100(y + 7)^2 - 4900 = -196 \] \[ 49(x + 2)^2 + 100(y + 7)^2 = 4900 \] \[ \frac{(x