The equation of the circle whose diameter lies on 2x + 3y = 3 and 16x - y = 4 and which passes through (4, 6) is

The equation of diameters of the circle are;
$$\begin{gathered} 2x+3y = 3 ...\left(\mathrm{i}\right) \\ 16\mathrm{x}-\mathrm{y} = 4 ...\left(\mathrm{ii}\right) \end{gathered}$$
Multiplying (i) by 8 and then subtracting (ii) from (i) we get;
$$\begin{gathered} 16x+24y-16x+y = 24-4 \\ \Rightarrow 25y = 20 \\ \Rightarrow y = \frac{20}{25} = \frac{4}{5} \end{gathered}$$
And from (ii) we have;
$$\begin{gathered} 16x-\frac{4}{5} = 4 \\ \Rightarrow 16x = 4+\frac{4}{5} \\ \Rightarrow 16x = \frac{24}{5} \\ \Rightarrow x = \frac{24}{5\times 16} = \frac{3}{10} \end{gathered}$$
So the centre of the circle is $\left(\frac{3}{10}, \frac{4}{5}\right)$
We know that the equation of the circle with the centre $\left(h,k\right)$ and radius r is ${\left(x-h\right)}^2+{\left(y-k\right)}^2 = r^2$
So the equation of the required circle with centre $\left(\frac{3}{10}, \frac{4}{5}\right)$ and radius r is ${\left(x-\frac{3}{10}\right)}^2+{\left(y-\frac{4}{5}\right)}^2 = r^2$
Since this circle passes through the point (4, 6), so we have;
$$\begin{gathered} {\left(4-\frac{3}{10}\right)}^2+{\left(6-\frac{4}{5}\right)}^2 = r^2 \\ \Rightarrow {\left(\frac{37}{10}\right)}^2+{\left(\frac{26}{5}\right)}^2 = r^2 \\ \Rightarrow r^2 = \frac{1369}{100}+\frac{676}{25} \\ \Rightarrow r^2 = \frac{4073}{100} ...\left(\mathrm{iii}\right) \end{gathered}$$
So the equation of the circle is given by;
$$\begin{gathered} {\left(x-\frac{3}{10}\right)}^2+{\left(y-\frac{4}{5}\right)}^2 = r^2 \\ \Rightarrow {\left(x-\frac{3}{10}\right)}^2+{\left(y-\frac{4}{5}\right)}^2 = \frac{4073}{100} \{\mathrm{using} (\mathrm{iii}\left)\right\} \\ \Rightarrow {\mathrm{x}}^2+\frac{9}{100}-\frac{6\mathrm{x}}{10}+{\mathrm{y}}^2+\frac{16}{25}-\frac{8\mathrm{y}}{5} = \frac{4073}{100} \\ \Rightarrow \frac{100{\mathrm{x}}^2+9-60\mathrm{x}+100{\mathrm{y}}^2+64-160\mathrm{y}}{100} = \frac{4073}{100} \\ \Rightarrow 100{\mathrm{x}}^2+100{\mathrm{y}}^2-60\mathrm{x}-160\mathrm{y}+73 = 4073 \\ \Rightarrow 100{\mathrm{x}}^2+100{\mathrm{y}}^2-60\mathrm{x}-160\mathrm{y} = 4000 \\ \Rightarrow 5{\mathrm{x}}^2+5{\mathrm{y}}^2-3\mathrm{x}-8\mathrm{y} = 200 \end{gathered}$$
Therefore the equation of the circle is $5x^2+5y^2-3x-8y = 200$