1. **Problem statement:** Find the length of the tangent from a given point to each circle. The length of the tangent from a point $(x_1,y_1)$ to a circle with equation $$Ax^2 + Ay^2 + Bx + Cy + D = 0$$ is given by $$\sqrt{S_1}$$ where $$S_1 = S(x_1,y_1) - r^2$$, with $$S(x,y) = x^2 + y^2 + \frac{B}{A}x + \frac{C}{A}y$$ and $$r^2 = \left(\frac{B}{2A}\right)^2 + \left(\frac{C}{2A}\right)^2 - \frac{D}{A}$$. 2. **First circle:** $$x^2 + y^2 + 5x + 4y - 20 = 0$$, point $(2,3)$. - Here, $A=1$, $B=5$, $C=4$, $D=-20$. - Calculate center: $$h = -\frac{B}{2A} = -\frac{5}{2} = -2.5$$, $$k = -\frac{C}{2A} = -2$$. - Calculate radius squared: $$r^2 = h^2 + k^2 - \frac{D}{A} = (-2.5)^2 + (-2)^2 - (-20) = 6.25 + 4 + 20 = 30.25$$. - Calculate $$S(2,3) = 2^2 + 3^2 + 5 \times 2 + 4 \times 3 = 4 + 9 + 10 + 12 = 35$$. - Length of tangent: $$\sqrt{S(2,3) - r^2} = \sqrt{35 - 30.25} = \sqrt{4.75} \approx 2.179$$. 3. **Second circle:** $$x^2 + y^2 - 3x + 2y - 10 = 0$$, point $(-4,1)$. - $A=1$, $B=-3$, $C=2$, $D=-10$. - Center: $$h = -\frac{-3}{2} = 1.5$$, $$k = -1$$. - Radius squared: $$r^2 = 1.5^2 + (-1)^2 - (-10) = 2.25 + 1 + 10 = 13.25$$. - Calculate $$S(-4,1) = (-4)^2 + 1^2 - 3 \times (-4) + 2 \times 1 = 16 + 1 + 12 + 2 = 31$$. - Length of tangent: $$\sqrt{31 - 13.25} = \sqrt{17.75} \approx 4.213$$. 4. **Third circle:** $$2x^2 + 2y^2 + x - 2y - 17 = 0$$, point $(5,-3)$. - Divide entire equation by 2: $$x^2 + y^2 + \frac{1}{2}x - y - \frac{17}{2} = 0$$. - $A=1$, $B=\frac{1}{2}$, $C=-1$, $D=-\frac{17}{2}$. - Center: $$h = -\frac{1/2}{2} = -\frac{1}{4} = -0.25$$, $$k = -\frac{-1}{2} = 0.5$$. - Radius squared: $$r^2 = (-0.25)^2 + 0.5^2 - \left(-\frac{17}{2}\right) = 0.0625 + 0.25 + 8.5 = 8.8125$$. - Calculate $$S(5,-3) = 5^2 + (-3)^2 + \frac{1}{2} \times 5 - 2 \times (-3) = 25 + 9 + 2.5 + 6 = 42.5$$. - Length of tangent: $$\sqrt{42.5 - 8.8125} = \sqrt{33.6875} \approx 5.805$$. 5. **Fourth circle:** $$3x^2 + 3y^2 + 5x + 6y - 30 = 0$$, point $(2,2)$. - Divide entire equation by 3: $$x^2 + y^2 + \frac{5}{3}x + 2y - 10 = 0$$. - $A=1$, $B=\frac{5}{3}$, $C=2$, $D=-10$. - Center: $$h = -\frac{5/3}{2} = -\frac{5}{6} \approx -0.8333$$, $$k = -1$$. - Radius squared: $$r^2 = (-0.8333)^2 + (-1)^2 - (-10) = 0.6944 + 1 + 10 = 11.6944$$. - Calculate $$S(2,2) = 2^2 + 2^2 + \frac{5}{3} \times 2 + 2 \times 2 = 4 + 4 + 3.3333 + 4 = 15.3333$$. - Length of tangent: $$\sqrt{15.3333 - 11.6944} = \sqrt{3.6389} \approx 1.908$$. **Final answers:** - Length of tangent to first circle: $\approx 2.179$ - Length of tangent to second circle: $\approx 4.213$ - Length of tangent to third circle: $\approx 5.805$ - Length of tangent to fourth circle: $\approx 1.908$