1. **State the problem:** Find the equation of a circle passing through points $(2,3)$ and $(-1,2)$ with its center on the line $2x - 3y = 0$. 2. **General form of circle equation:** A circle with center $(h,k)$ and radius $r$ is given by: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. **Condition for center:** The center $(h,k)$ lies on the line $2x - 3y = 0$, so: $$ 2h - 3k = 0 \implies 2h = 3k \implies h = \frac{3k}{2} $$ 4. **Circle passes through $(2,3)$:** $$ (2 - h)^2 + (3 - k)^2 = r^2 $$ 5. **Circle passes through $(-1,2)$:** $$ (-1 - h)^2 + (2 - k)^2 = r^2 $$ 6. **Set the two expressions for $r^2$ equal:** $$ (2 - h)^2 + (3 - k)^2 = (-1 - h)^2 + (2 - k)^2 $$ 7. **Substitute $h = \frac{3k}{2}$:** $$ \left(2 - \frac{3k}{2}\right)^2 + (3 - k)^2 = \left(-1 - \frac{3k}{2}\right)^2 + (2 - k)^2 $$ 8. **Expand each term:** $$ \left(2 - \frac{3k}{2}\right)^2 = \left(\frac{4}{2} - \frac{3k}{2}\right)^2 = \left(\frac{4 - 3k}{2}\right)^2 = \frac{(4 - 3k)^2}{4} $$ $$ (3 - k)^2 = 9 - 6k + k^2 $$ $$ \left(-1 - \frac{3k}{2}\right)^2 = \left(-\frac{2}{2} - \frac{3k}{2}\right)^2 = \left(-\frac{2 + 3k}{2}\right)^2 = \frac{(2 + 3k)^2}{4} $$ $$ (2 - k)^2 = 4 - 4k + k^2 $$ 9. **Rewrite the equation:** $$ \frac{(4 - 3k)^2}{4} + 9 - 6k + k^2 = \frac{(2 + 3k)^2}{4} + 4 - 4k + k^2 $$ 10. **Multiply both sides by 4 to clear denominators:** $$ (4 - 3k)^2 + 4(9 - 6k + k^2) = (2 + 3k)^2 + 4(4 - 4k + k^2) $$ 11. **Expand:** $$ (4 - 3k)^2 = 16 - 24k + 9k^2 $$ $$ 4(9 - 6k + k^2) = 36 - 24k + 4k^2 $$ $$ (2 + 3k)^2 = 4 + 12k + 9k^2 $$ $$ 4(4 - 4k + k^2) = 16 - 16k + 4k^2 $$ 12. **Sum left side:** $$ 16 - 24k + 9k^2 + 36 - 24k + 4k^2 = 52 - 48k + 13k^2 $$ 13. **Sum right side:** $$ 4 + 12k + 9k^2 + 16 - 16k + 4k^2 = 20 - 4k + 13k^2 $$ 14. **Set equal and simplify:** $$ 52 - 48k + 13k^2 = 20 - 4k + 13k^2 $$ 15. **Cancel $13k^2$ on both sides:** $$ 52 - 48k = 20 - 4k $$ 16. **Rearrange:** $$ 52 - 20 = -4k + 48k \implies 32 = 44k $$ 17. **Solve for $k$:** $$ k = \frac{32}{44} = \frac{8}{11} $$ 18. **Find $h$ using $h = \frac{3k}{2}$:** $$ h = \frac{3}{2} \times \frac{8}{11} = \frac{24}{22} = \frac{12}{11} $$ 19. **Find radius squared $r^2$ using point $(2,3)$:** $$ r^2 = (2 - h)^2 + (3 - k)^2 = \left(2 - \frac{12}{11}\right)^2 + \left(3 - \frac{8}{11}\right)^2 $$ $$ = \left(\frac{22}{11} - \frac{12}{11}\right)^2 + \left(\frac{33}{11} - \frac{8}{11}\right)^2 = \left(\frac{10}{11}\right)^2 + \left(\frac{25}{11}\right)^2 $$ $$ = \frac{100}{121} + \frac{625}{121} = \frac{725}{121} $$ 20. **Final equation of the circle:** $$ \left(x - \frac{12}{11}\right)^2 + \left(y - \frac{8}{11}\right)^2 = \frac{725}{121} $$ This is the required circle equation.