1. **Problem statement:** Find the equation of the tangent to the circle $x^2 + y^2 + 2x - 3y + 13 = 0$ at the point $(1, -2)$. 2. **Formula and rules:** The equation of the tangent to a circle $x^2 + y^2 + 2gx + 2fy + c = 0$ at point $(x_1, y_1)$ on the circle is given by: $$xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$$ where $g$, $f$, and $c$ come from the circle equation. 3. **Identify parameters:** Rewrite the circle equation as $x^2 + y^2 + 2x - 3y + 13 = 0$. Here, $2g = 2 \Rightarrow g = 1$, $2f = -3 \Rightarrow f = -\frac{3}{2}$, and $c = 13$. 4. **Apply formula:** Substitute $x_1 = 1$, $y_1 = -2$, $g = 1$, $f = -\frac{3}{2}$, and $c = 13$ into the tangent formula: $$x(1) + y(-2) + 1(x + 1) - \frac{3}{2}(y - 2) + 13 = 0$$ 5. **Simplify:** $$x - 2y + x + 1 - \frac{3}{2}y + 3 + 13 = 0$$ $$2x - 2y - \frac{3}{2}y + 17 = 0$$ 6. **Combine like terms:** $$2x - \left(2 + \frac{3}{2}\right)y + 17 = 0$$ $$2x - \frac{7}{2}y + 17 = 0$$ 7. **Clear fractions by multiplying both sides by 2:** $$2 \times \left(2x - \frac{7}{2}y + 17\right) = 0 \times 2$$ $$\cancel{2} \times 2x - \cancel{2} \times \frac{7}{2}y + \cancel{2} \times 17 = 0$$ $$4x - 7y + 34 = 0$$ --- 1. **Problem statement:** Find the equation of the tangent to the circle $x^2 + y^2 - 2x - y - 15 = 0$ at the point $(-1, -3)$. 2. **Identify parameters:** Rewrite the circle equation as $x^2 + y^2 + 2gx + 2fy + c = 0$. Here, $2g = -2 \Rightarrow g = -1$, $2f = -1 \Rightarrow f = -\frac{1}{2}$, and $c = -15$. 3. **Apply formula:** Substitute $x_1 = -1$, $y_1 = -3$, $g = -1$, $f = -\frac{1}{2}$, and $c = -15$ into the tangent formula: $$x(-1) + y(-3) -1(x - 1) - \frac{1}{2}(y - 3) - 15 = 0$$ 4. **Simplify:** $$-x - 3y - x + 1 - \frac{1}{2}y + \frac{3}{2} - 15 = 0$$ $$-2x - 3y - \frac{1}{2}y + 1 + \frac{3}{2} - 15 = 0$$ 5. **Combine like terms:** $$-2x - \left(3 + \frac{1}{2}\right)y + \left(1 + \frac{3}{2} - 15\right) = 0$$ $$-2x - \frac{7}{2}y - \frac{21}{2} = 0$$ 6. **Clear fractions by multiplying both sides by 2:** $$2 \times \left(-2x - \frac{7}{2}y - \frac{21}{2}\right) = 0 \times 2$$ $$\cancel{2} \times -2x - \cancel{2} \times \frac{7}{2}y - \cancel{2} \times \frac{21}{2} = 0$$ $$-4x - 7y - 21 = 0$$ **Final answers:** - Tangent at $(1, -2)$: $$4x - 7y + 34 = 0$$ - Tangent at $(-1, -3)$: $$-4x - 7y - 21 = 0$$