1. **Problem Statement:** Write the general equation of a circle with center at (2, 6) and radius 9 units, then find values of D, E, and F. 2. **Formula:** The standard form of a circle's equation is $$(x - h)^2 + (y - k)^2 = r^2$$ where $(h, k)$ is the center and $r$ is the radius. 3. **Step 1:** Substitute $h=2$, $k=6$, and $r=9$: $$ (x - 2)^2 + (y - 6)^2 = 9^2 $$ 4. **Step 2:** Expand the squares: $$ (x^2 - 4x + 4) + (y^2 - 12y + 36) = 81 $$ 5. **Step 3:** Combine like terms: $$ x^2 + y^2 - 4x - 12y + 40 = 81 $$ 6. **Step 4:** Bring all terms to one side: $$ x^2 + y^2 - 4x - 12y + 40 - 81 = 0 $$ $$ x^2 + y^2 - 4x - 12y - 41 = 0 $$ 7. **Step 5:** Identify $D$, $E$, and $F$ from the general form $$x^2 + y^2 + Dx + Ey + F = 0$$: $$ D = -4, \quad E = -12, \quad F = -41 $$ --- Repeat the same steps for the other centers and radii: **2. Center (-7, 2), radius 15:** $$(x + 7)^2 + (y - 2)^2 = 15^2$$ $$ (x^2 + 14x + 49) + (y^2 - 4y + 4) = 225 $$ $$ x^2 + y^2 + 14x - 4y + 53 = 225 $$ $$ x^2 + y^2 + 14x - 4y + 53 - 225 = 0 $$ $$ x^2 + y^2 + 14x - 4y - 172 = 0 $$ $$ D = 14, \quad E = -4, \quad F = -172 $$ **3. Center (3, 5), radius 8:** $$(x - 3)^2 + (y - 5)^2 = 8^2$$ $$ (x^2 - 6x + 9) + (y^2 - 10y + 25) = 64 $$ $$ x^2 + y^2 - 6x - 10y + 34 = 64 $$ $$ x^2 + y^2 - 6x - 10y + 34 - 64 = 0 $$ $$ x^2 + y^2 - 6x - 10y - 30 = 0 $$ $$ D = -6, \quad E = -10, \quad F = -30 $$ **4. Center (-1, -4), radius 6:** $$(x + 1)^2 + (y + 4)^2 = 6^2$$ $$ (x^2 + 2x + 1) + (y^2 + 8y + 16) = 36 $$ $$ x^2 + y^2 + 2x + 8y + 17 = 36 $$ $$ x^2 + y^2 + 2x + 8y + 17 - 36 = 0 $$ $$ x^2 + y^2 + 2x + 8y - 19 = 0 $$ $$ D = 2, \quad E = 8, \quad F = -19 $$ **Summary:** - Problem 1: $x^2 + y^2 - 4x - 12y - 41 = 0$, $D=-4$, $E=-12$, $F=-41$ - Problem 2: $x^2 + y^2 + 14x - 4y - 172 = 0$, $D=14$, $E=-4$, $F=-172$ - Problem 3: $x^2 + y^2 - 6x - 10y - 30 = 0$, $D=-6$, $E=-10$, $F=-30$ - Problem 4: $x^2 + y^2 + 2x + 8y - 19 = 0$, $D=2$, $E=8$, $F=-19$ Each equation represents a circle with the given center and radius, expressed in the general form $x^2 + y^2 + Dx + Ey + F = 0$.