​HELP ME WITH THE 4 QUEST ASAP

Dear student

Two circles S = x² +y² +2gx +2 f y +c = 0 and

S' = x² +y² +2 g' x +2 f ' y +c' = 0 cut orthogonally if 2 (g g' +f f') = c +c'.

So, in this case,

$$\begin{gathered} 2\left(1\times 0 + \mathrm{k}\times \mathrm{k}\right) = \mathrm{k} + 6 \\ \Rightarrow 2{\mathrm{k}}^2 - \mathrm{k} - 6 = 0 \\ \Rightarrow 2{\mathrm{k}}^2 - 4\mathrm{k} + 3\mathrm{k} - 6 = 0 \\ \Rightarrow 2\mathrm{k}\left(\mathrm{k} - 2\right) + 3\left(\mathrm{k} - 2\right) = 0 \\ \Rightarrow \left(2\mathrm{k}+3\right)\left(\mathrm{k}-2\right) = 0 \\ \Rightarrow 2\mathrm{k}+3 = 0 \mathrm{or} \mathrm{k}-2 = 0 \\ \Rightarrow \mathrm{k} = -\frac{3}{2} \mathrm{or} \mathrm{k} = 2 \end{gathered}$$

Regards