1. **Problem statement:** Find the equation of a circle passing through points $(1,2)$ and $(3,4)$ and tangent to the line $3x + y - 3 = 0$. 2. **General form of a circle:** The equation of a circle can be written as $$x^2 + y^2 + 2gx + 2fy + c = 0,$$ where the center is $(-g, -f)$ and radius is $r = \sqrt{g^2 + f^2 - c}$. 3. **Conditions:** - The circle passes through $(1,2)$ and $(3,4)$, so these points satisfy the circle equation: $$1^2 + 2^2 + 2g(1) + 2f(2) + c = 0 \implies 1 + 4 + 2g + 4f + c = 0,$$ $$3^2 + 4^2 + 2g(3) + 2f(4) + c = 0 \implies 9 + 16 + 6g + 8f + c = 0.$$ - The circle is tangent to the line $3x + y - 3 = 0$. The distance from the center $(-g,-f)$ to the line equals the radius: $$\frac{|3(-g) + (-f) - 3|}{\sqrt{3^2 + 1^2}} = r = \sqrt{g^2 + f^2 - c}.$$ 4. **Simplify the point conditions:** From the first point: $$5 + 2g + 4f + c = 0 \implies 2g + 4f + c = -5,$$ From the second point: $$25 + 6g + 8f + c = 0 \implies 6g + 8f + c = -25.$$ 5. **Subtract the first from the second:** $$(6g + 8f + c) - (2g + 4f + c) = -25 - (-5) \implies 4g + 4f = -20 \implies g + f = -5.$$ 6. **Express $c$ from the first equation:** $$c = -5 - 2g - 4f.$$ 7. **Use the tangent condition:** $$\frac{| -3g - f - 3 |}{\sqrt{10}} = \sqrt{g^2 + f^2 - c}.$$ Square both sides: $$\frac{(-3g - f - 3)^2}{10} = g^2 + f^2 - c.$$ Substitute $c$: $$\frac{(-3g - f - 3)^2}{10} = g^2 + f^2 + 5 + 2g + 4f.$$ 8. **Expand and simplify:** $$(-3g - f - 3)^2 = 9g^2 + 6gf + f^2 + 18g + 6f + 9,$$ so $$\frac{9g^2 + 6gf + f^2 + 18g + 6f + 9}{10} = g^2 + f^2 + 5 + 2g + 4f.$$ Multiply both sides by 10: $$9g^2 + 6gf + f^2 + 18g + 6f + 9 = 10g^2 + 10f^2 + 50 + 20g + 40f.$$ 9. **Bring all terms to one side:** $$0 = 10g^2 + 10f^2 + 50 + 20g + 40f - 9g^2 - 6gf - f^2 - 18g - 6f - 9,$$ $$0 = (10g^2 - 9g^2) + (10f^2 - f^2) + (20g - 18g) + (40f - 6f) + 50 - 9 - 6gf,$$ $$0 = g^2 + 9f^2 + 2g + 34f + 41 - 6gf.$$ 10. **Recall from step 5:** $g + f = -5$, so $f = -5 - g$. 11. **Substitute $f$ into the equation:** $$g^2 + 9(-5 - g)^2 + 2g + 34(-5 - g) + 41 - 6g(-5 - g) = 0.$$ Expand: $$g^2 + 9(g^2 + 10g + 25) + 2g - 170 - 34g + 41 + 30g + 6g^2 = 0,$$ $$g^2 + 9g^2 + 90g + 225 + 2g - 170 - 34g + 41 + 30g + 6g^2 = 0,$$ Combine like terms: $$(1 + 9 + 6)g^2 + (90 + 2 - 34 + 30)g + (225 - 170 + 41) = 0,$$ $$16g^2 + 88g + 96 = 0.$$ 12. **Divide entire equation by 8:** $$2g^2 + 11g + 12 = 0.$$ 13. **Solve quadratic:** $$g = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 2 \cdot 12}}{2 \cdot 2} = \frac{-11 \pm \sqrt{121 - 96}}{4} = \frac{-11 \pm 5}{4}.$$ 14. **Two solutions for $g$:** $$g_1 = \frac{-11 + 5}{4} = -\frac{3}{2}, \quad g_2 = \frac{-11 - 5}{4} = -4.$$ 15. **Find corresponding $f$ values:** $$f = -5 - g,$$ $$f_1 = -5 - (-\frac{3}{2}) = -\frac{7}{2}, \quad f_2 = -5 - (-4) = -1.$$ 16. **Find $c$ for each pair:** $$c = -5 - 2g - 4f,$$ For $(g_1, f_1)$: $$c = -5 - 2(-\frac{3}{2}) - 4(-\frac{7}{2}) = -5 + 3 + 14 = 12,$$ For $(g_2, f_2)$: $$c = -5 - 2(-4) - 4(-1) = -5 + 8 + 4 = 7.$$ 17. **Write the two circle equations:** $$x^2 + y^2 + 2(-\frac{3}{2})x + 2(-\frac{7}{2})y + 12 = 0 \implies x^2 + y^2 - 3x - 7y + 12 = 0,$$ $$x^2 + y^2 + 2(-4)x + 2(-1)y + 7 = 0 \implies x^2 + y^2 - 8x - 2y + 7 = 0.$$ **Final answer:** The two possible circle equations are $$x^2 + y^2 - 3x - 7y + 12 = 0$$ and $$x^2 + y^2 - 8x - 2y + 7 = 0.$$