Kwadraatafsplitsen geeft \(x^{2} + (y - 4)^{2} = 20\)
Dus \(M (0 , 4)\) en \(r = \sqrt{20} \text{.}\)

De lijn \(m\) door \(M\) en \(A\) heeft \(\text{rc}_{m} = {\Delta y \over \Delta x} = {4 - 6 \over 0 - 4} = \frac{1}{2} \text{.}\)

\(\begin{rcases}l \perp m \text{, dus } \text{rc}_{l} ⋅ \text{rc}_{m} = -1 \\ \text{rc}_{m} = \frac{1}{2}\end{rcases} \text{rc}_{l} = -2\)

\(\begin{rcases}y = -2 x + b \\ \text{door } A (4 , 6)\end{rcases} \begin{matrix}6 = -2 ⋅ 4 + b \\ 6 = -8 + b \\ b = 14\end{matrix}\)
Dus \(l{:}\,y = -2 x + 14 \text{.}\)

\(\begin{rcases}c{:}\,x^{2} + y^{2} + 4 y - 1 = 0 \\ x = 2\end{rcases}\) geeft
\(2^{2} + y^{2} + 0 ⋅ 2 + 4 y - 1 = 0\)
\(y^{2} + 4 y + 3 = 0\)
\((y + 3) (y + 1) = 0\)
\(y = -3 ∨ y = -1\)

\(y_{A} > y_{B} \text{,}\) dus \(A (2 , -1)\) en \(B (2 , -3) \text{.}\)

(kwadraatafsplitsen)
\(x^{2} + (y + 2)^{2} - 4 - 1 = 0\)
\(x^{2} + (y + 2)^{2} = 5\)
Dus \(M (0 , -2) \text{.}\)

(voor de lijn door \(B\) en \(M\) geldt)
\(\text{rc}_{\text{BM}} = {\Delta y \over \Delta x} = {-2 - -3 \over 0 - 2} = -\frac{1}{2} \text{.}\)

\(\begin{rcases}\text{BM} \perp l \text{, dus } \text{rc}_{\text{BM}} ⋅ \text{rc}_{l} = -1 \\ \text{rc}_{\text{BM}} = -\frac{1}{2}\end{rcases} \text{rc}_{l} = 2\)

\(\begin{rcases}l{:}\,y = 2 x + b \\ \text{door } B (2 , -3)\end{rcases} \begin{matrix}-3 = 2 ⋅ 2 + b \\ -3 = 4 + b \\ b = -7\end{matrix}\)
Dus \(l{:}\,y = 2 x - 7 \text{.}\)

\(\begin{rcases}l{:}\,y = a x + b \\ A (15 , 15)\end{rcases} \begin{matrix}a ⋅ 15 + b = 15 \\ b = 15-15 a\end{matrix}\)
oftewel
\(l \text{: } y = a x+15-15 a\)
\(l \text{: } a x-y+15-15 a = 0\)

Van \(c\) is \(M (8 , 6)\) en \(r = 3 \text{.}\)
\(d(M , l) = \frac{\vert 8 a-6+15-15 a \vert}{\sqrt{a^{2}+(-1)^{2}}} = \frac{\vert -7 a+9 \vert}{\sqrt{a^{2}+1}} = 3\)

(oplossen)
\(\vert -7 a+9 \vert = 3 \sqrt{a^{2}+1}\)
\((\vert -7 a+9 \vert)^{2} = (3 \sqrt{a^{2}+1})^{2}\)
\(49 a^{2}-126 a+81 = 9 a^{2}+9\)

\(40 a^{2} - 126 a + 72 = 0\)
\(20 a^{2} - 63 a + 36 = 0\)
\(D = (-63)^{2} - 4 ⋅ 20 ⋅ 36 = 1\,089\)
\(a = {63 - \sqrt{1\,089} \over 2 ⋅ 20} ∨ a = {63 + \sqrt{1\,089} \over 2 ⋅ 20}\)
\(a = \frac{3}{4} ∨ a = 2\frac{2}{5}\)

\(a = \frac{3}{4}\) geeft \(b = 15-15 \cdot {3 \over 4} = 3\frac{3}{4} \text{,}\) dus \(l_{1} \text{: } y = {3 \over 4} x+3\frac{3}{4}\)
\(a = 2\frac{2}{5}\) geeft \(b = 15-15 \cdot 2\frac{2}{5} = -21 \text{,}\) dus \(l_{2} \text{: } y = 2\frac{2}{5} x-21\)

\(l \text{: } y = x+b\) geeft \(l \text{: } x-y+b = 0\)

(kwadraatafsplitsen)
\((x-6)^{2}-36+y^{2}+34 = 0\)
\((x - 6)^{2} + y^{2} = 2\)
Dus \(M (6 , 0)\) en \(r = \sqrt{2} \text{.}\)

\(d(M , l) = \frac{\vert 1 \cdot 6-1 \cdot 0+b \vert}{\sqrt{1^{2}+(-1)^{2}}} = \frac{\vert 6+b \vert}{\sqrt{2}} = \sqrt{2}\)

(oplossen)
\(\vert 6+b \vert = \sqrt{2} \cdot \sqrt{2} = 2\)
\(6+b = 2 ∨ 6+b = -2\)
\(b = -6+2 = -4 ∨ b = -6-2 = -8\)

\(b = -4\) geeft \(l_{1} \text{: } y = x-4\)
\(b = -8\) geeft \(l_{2} \text{: } y = x-8\)