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+## Analytic Geometry Part 2
+
+### Question 3
+
+Shortest distance = straight perpendicular line that connets $`A`$ to a point on line $`\overline{GH}`$
+
+$`M_{GH} = \dfrac{42+30}{38 + 16} = \dfrac{72}{54} = \dfrac{4}{3}`$
+
+$`M_{\perp GH} = \dfrac{-3}{4}`$
+
+$`y_{\perp GH} - 32 = \dfrac{-3}{4}(x+16)`$
+
+$`y_{\perp GH} = \dfrac{-3}{4}x + 20 \quad (1)`$
+
+$`y_{GH} + 30 = \dfrac{4}{3}(x+16)`$
+
+$`y_{GH} = \dfrac{4}{3}x - \dfrac{26}{3} \quad (2)`$
+
+```math
+\begin{cases}
+
+y_{\perp GH} = \dfrac{-3}{4}x + 20 & \text{(1)} \\
+
+\\
+
+y_{GH} = \dfrac{4}{3}x - \dfrac{26}{3} & \text{(2}) \\
+\end{cases}
+```
+
+Sub $`(1)`$ into $`(2)`$
+
+$`\dfrac{-3}{4}x + 20 = \dfrac{4}{3}x - \dfrac{26}{3}`$
+
+$`-9x + (12)20 = 16x - 4(26)`$
+
+$`25x = 344`$
+
+$`x = \dfrac{344}{25} \quad (3)`$
+
+Sub $`(3)`$ into $`(1)`$
+
+$`y = \dfrac{-3}{4}(\dfrac{344}{25}) + 20`$
+
+$`y = \dfrac{-258}{25} + 20`$
+
+$`y = \dfrac{-258}{25} + \dfrac{500}{25}`$
+
+$`y = \dfrac{242}{25}`$
+
+Distance $`= \sqrt{(-16-\dfrac{344}{25})^2 + (32 - \dfrac{242}{25})^2} = 37.2`$
+
+$`\therefore`$ The shortest length pipe is $`37.2`$ units.
+
+
+### Question 4
+
+Let $`(x, y)`$ be the center of the circle, and $`r`$ be the radius of the circle.
+
+
+```math
+\begin{cases}
+(x-4)^2 + (y-8)^2 = r^2 & \text{(1)} \\
+
+(x-5)^2 + (y-1)^2 = r^2 & \text{(2)} \\
+
+(x+2)^2 + y^2 = r^2 & \text{(3)} \\
+
+\end{cases}
+```
+
+Sub $`(1)`$ into $`(2)`$
+
+$`x^2 - 8x + 16 + y^2 - 16y + 64 = x^2 - 10x + 25 + y^2 -2y + 1`$
+
+$`-8x -16y + 80 = -10x - 2y + 26`$
+
+$`2x - 14y = -54`$
+
+$`x - 7y = -27 \quad (4)`$
+
+Sub $`(2)`$ into $`(3)`$
+
+$`x^2 - 10x + 25 + y^2 - 2y + 1 = x^2 + 4x + 4 + y^2`$
+
+$`-10x - 2y +26 = 4x + 4`$
+
+$`14x + 2y = 22`$
+
+$`7x + y = 11`$
+
+$`y = 11 - 7x \quad (5)`$
+
+Sub $`(5)`$ into $`(4)`$
+
+$`x - 7(11-7x) = -27`$
+
+$`x - 77+ 49x = 27`$
+
+$`50x = 50`$
+
+$`x = 1 \quad (6)`$ 
+
+Sub $`(6)`$ into $`(5)`$
+
+$`y = 11 - 7(1)`$
+
+$`y = 4 \quad (7)`$
+
+Sub $`(6), (7)`$ into $`(3)`$
+
+$`(1+2)^2 + 4^2 = r^2`$
+
+$`r^2 = 16 + 9`$
+
+$`r^2 = 25`$
+
+$`\therefore (x-1)^2 + (y-4)^2 = 25`$ is the equation of the circle.
+