1. **Problem:** Find the canonical form of the surface given by $$4x^2 + 9y^2 + 25z^2 = 900$$ and identify its type. 2. **Formula and rules:** The general form of an ellipsoid is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$ where $a,b,c$ are the semi-axes lengths. 3. **Work:** Divide both sides by 900: $$\frac{4x^2}{900} + \frac{9y^2}{900} + \frac{25z^2}{900} = 1$$ Simplify: $$\frac{x^2}{225} + \frac{y^2}{100} + \frac{z^2}{36} = 1$$ 4. **Interpretation:** This is an ellipsoid centered at the origin with semi-axes $a=15$, $b=10$, $c=6$. --- 1. **Problem:** Find the canonical form of $$16x^2 + 4y^2 + z^2 - 64 = 0$$. 2. **Work:** Rewrite as: $$16x^2 + 4y^2 + z^2 = 64$$ Divide both sides by 64: $$\frac{16x^2}{64} + \frac{4y^2}{64} + \frac{z^2}{64} = 1$$ Simplify: $$\frac{x^2}{4} + \frac{y^2}{16} + \frac{z^2}{64} = 1$$ 3. **Interpretation:** Ellipsoid centered at origin with semi-axes $a=2$, $b=4$, $c=8$. --- 1. **Problem:** Canonical form of $$9x^2 + 36y^2 + 49z^2 = 441$$. 2. **Work:** Divide both sides by 441: $$\frac{9x^2}{441} + \frac{36y^2}{441} + \frac{49z^2}{441} = 1$$ Simplify: $$\frac{x^2}{49} + \frac{y^2}{12.25} + \frac{z^2}{9} = 1$$ 3. **Interpretation:** Ellipsoid with semi-axes $a=7$, $b=3.5$, $c=3$. --- 1. **Problem:** Canonical form of $$x^2 + 2y^2 + 3z^2 = 6$$. 2. **Work:** Divide both sides by 6: $$\frac{x^2}{6} + \frac{2y^2}{6} + \frac{3z^2}{6} = 1$$ Simplify: $$\frac{x^2}{6} + \frac{y^2}{3} + \frac{z^2}{2} = 1$$ 3. **Interpretation:** Ellipsoid with semi-axes $a=\sqrt{6}$, $b=\sqrt{3}$, $c=\sqrt{2}$. --- 1. **Problem:** Canonical form of $$4x^2 + y^2 + 25z^2 = 900$$. 2. **Work:** Divide both sides by 900: $$\frac{4x^2}{900} + \frac{y^2}{900} + \frac{25z^2}{900} = 1$$ Simplify: $$\frac{x^2}{225} + \frac{y^2}{900} + \frac{z^2}{36} = 1$$ 3. **Interpretation:** Ellipsoid with semi-axes $a=15$, $b=30$, $c=6$. --- 1. **Problem:** Canonical form of $$25x^2 + 9y^2 + 16z^2 = 400$$. 2. **Work:** Divide both sides by 400: $$\frac{25x^2}{400} + \frac{9y^2}{400} + \frac{16z^2}{400} = 1$$ Simplify: $$\frac{x^2}{16} + \frac{y^2}{44.44} + \frac{z^2}{25} = 1$$ 3. **Interpretation:** Ellipsoid with semi-axes $a=4$, $b=\sqrt{44.44}$, $c=5$. --- 1. **Problem:** Canonical form of $$x^2 + y^2 + 4z^2 - 9 = 0$$. 2. **Work:** Rewrite: $$x^2 + y^2 + 4z^2 = 9$$ Divide both sides by 9: $$\frac{x^2}{9} + \frac{y^2}{9} + \frac{4z^2}{9} = 1$$ Simplify: $$\frac{x^2}{9} + \frac{y^2}{9} + \frac{z^2}{2.25} = 1$$ 3. **Interpretation:** Ellipsoid with semi-axes $a=3$, $b=3$, $c=1.5$. --- 1. **Problem:** Canonical form of $$49x^2 + 16y^2 + 9z^2 = 784$$. 2. **Work:** Divide both sides by 784: $$\frac{49x^2}{784} + \frac{16y^2}{784} + \frac{9z^2}{784} = 1$$ Simplify: $$\frac{x^2}{16} + \frac{y^2}{49} + \frac{z^2}{87.11} = 1$$ 3. **Interpretation:** Ellipsoid with semi-axes $a=4$, $b=7$, $c=\sqrt{87.11}$. --- 1. **Problem:** Canonical form of $$4x^2 + 4y^2 + z^2 = 36$$. 2. **Work:** Divide both sides by 36: $$\frac{4x^2}{36} + \frac{4y^2}{36} + \frac{z^2}{36} = 1$$ Simplify: $$\frac{x^2}{9} + \frac{y^2}{9} + \frac{z^2}{36} = 1$$ 3. **Interpretation:** Ellipsoid with semi-axes $a=3$, $b=3$, $c=6$. --- 1. **Problem:** Canonical form of shifted ellipsoid $$9(x-1)^2 + 4(y+2)^2 + (z-3)^2 = 6$$. 2. **Work:** Divide both sides by 6: $$\frac{9(x-1)^2}{6} + \frac{4(y+2)^2}{6} + \frac{(z-3)^2}{6} = 1$$ Simplify: $$\frac{(x-1)^2}{\frac{2}{3}} + \frac{(y+2)^2}{\frac{3}{2}} + \frac{(z-3)^2}{6} = 1$$ 3. **Interpretation:** Ellipsoid centered at $(1,-2,3)$ with semi-axes $a=\sqrt{\frac{2}{3}}$, $b=\sqrt{\frac{3}{2}}$, $c=\sqrt{6}$. --- 1. **Problem:** Canonical form of $$4x^2 + 9y^2 + 25z^2 - 200 = 0$$. 2. **Work:** Rewrite: $$4x^2 + 9y^2 + 25z^2 = 200$$ Divide both sides by 200: $$\frac{4x^2}{200} + \frac{9y^2}{200} + \frac{25z^2}{200} = 1$$ Simplify: $$\frac{x^2}{50} + \frac{y^2}{\frac{200}{9}} + \frac{z^2}{8} = 1$$ 3. **Interpretation:** Ellipsoid with semi-axes $a=\sqrt{50}$, $b=\sqrt{\frac{200}{9}}$, $c=\sqrt{8}$. --- **Summary:** All given equations represent ellipsoids (some shifted) with canonical forms found by dividing through by the constant on the right side and rewriting in the form $$\frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2} + \frac{(z-z_0)^2}{c^2} = 1$$. **Graphical note:** Each ellipsoid is centered at the origin except equation 10, which is shifted to $(1,-2,3)$.