Write each equation in standard form. Then identify the
conic as an ellipse or a circle.
11.
12.
13.
14.

Ask by Munoz Rodriquez. in the United States

Mar 31,2025

Question

Write each equation in standard form. Then identify the
conic as an ellipse or a circle.
11.
12.
13.
14.

Ask by Munoz Rodriquez. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

1. **Equation:** $$ 16x^2 + 4y^2 = 64 $$

- **Step 1:** Divide both sides by 64 to simplify:
     
     $$
     \frac{16x^2}{64} + \frac{4y^2}{64} = 1 \quad \Longrightarrow \quad \frac{x^2}{4} + \frac{y^2}{16} = 1.
     $$
     
   - **Standard Form:** 
     
     $$
     \frac{x^2}{4} + \frac{y^2}{16} = 1.
     $$
     
   - **Conic Identification:**  
     Since the equation has both $$ x^2 $$ and $$ y^2 $$ terms with different denominators and positive signs, it represents an **ellipse**.

2. **Equation:** $$ x^2 + y^2 - 6x - 2y = -1 $$

- **Step 1:** Group the $$x$$ and $$y$$ terms:
     
     $$
     (x^2 - 6x) + (y^2 - 2y) = -1.
     $$
     
   - **Step 2:** Complete the square for the $$x$$-terms.  
     For $$ x^2 - 6x $$, take half of $$-6$$ (i.e. $$-3$$) and square it:
     
     $$
     x^2 - 6x = (x - 3)^2 - 9.
     $$
     
   - **Step 3:** Complete the square for the $$y$$-terms.  
     For $$ y^2 - 2y $$, take half of $$-2$$ (i.e. $$-1$$) and square it:
     
     $$
     y^2 - 2y = (y - 1)^2 - 1.
     $$
     
   - **Step 4:** Substitute these back into the equation:
     
     $$
     (x - 3)^2 - 9 + (y - 1)^2 - 1 = -1.
     $$
     
   - **Step 5:** Combine constant terms:
     
     $$
     (x - 3)^2 + (y - 1)^2 - 10 = -1 \quad \Longrightarrow \quad (x - 3)^2 + (y - 1)^2 = 9.
     $$
     
   - **Standard Form:** 
     
     $$
     (x - 3)^2 + (y - 1)^2 = 9.
     $$
     
   - **Conic Identification:**  
     This is the standard form of a **circle** with center $$(3, 1)$$ and radius $$3$$.

3. **Equation:** $$ 25x^2 + 9y^2 - 54y = 144 $$

- **Step 1:** Group terms and isolate the $$y$$ terms for completion:
     
     $$
     25x^2 + \left(9y^2 - 54y\right) = 144.
     $$
     
   - **Step 2:** Complete the square for the $$y$$-terms.  
     Factor out 9:
     
     $$
     9\left(y^2 - 6y\right).
     $$
     
     Take half of $$-6$$ (i.e. $$-3$$) and square it:
     
     $$
     y^2 - 6y = (y - 3)^2 - 9.
     $$
     
     Thus:
     
     $$
     9\left(y^2 - 6y\right) = 9\left((y - 3)^2 - 9\right) = 9(y - 3)^2 - 81.
     $$
     
   - **Step 3:** Substitute back into the equation:
     
     $$
     25x^2 + 9(y - 3)^2 - 81 = 144.
     $$
     
   - **Step 4:** Add 81 to both sides:
     
     $$
     25x^2 + 9(y - 3)^2 = 225.
     $$
     
   - **Step 5:** Divide both sides by 225:
     
     $$
     \frac{25x^2}{225} + \frac{9(y - 3)^2}{225} = 1 \quad \Longrightarrow \quad \frac{x^2}{9} + \frac{(y - 3)^2}{25} = 1.
     $$
     
   - **Standard Form:** 
     
     $$
     \frac{x^2}{9} + \frac{(y - 3)^2}{25} = 1.
     $$
     
   - **Conic Identification:**  
     This is the equation of an **ellipse**.

4. **Equation:** $$ 16x^2 + 9y^2 + 128x - 54y = -193 $$

- **Step 1:** Group the $$x$$ terms and the $$y$$ terms:
     
     $$
     (16x^2 + 128x) + (9y^2 - 54y) = -193.
     $$
     
   - **Step 2:** Complete the square for the $$x$$-terms.  
     Factor out 16:
     
     $$
     16\left(x^2 + 8x\right).
     $$
     
     Take half of $$8$$ (i.e. $$4$$) and square it:
     
     $$
     x^2 + 8x = (x + 4)^2 - 16.
     $$
     
     Thus:
     
     $$
     16(x^2 + 8x) = 16\left((x + 4)^2 - 16\right) = 16(x + 4)^2 - 256.
     $$
     
   - **Step 3:** Complete the square for the $$y$$-terms.  
     Factor out 9:
     
     $$
     9\left(y^2 - 6y\right).
     $$
     
     Take half of $$-6$$ (i.e. $$-3$$) and square it:
     
     $$
     y^2 - 6y = (y - 3)^2 - 9.
     $$
     
     Thus:
     
     $$
     9(y^2 - 6y) = 9\left((y - 3)^2 - 9\right) = 9(y - 3)^2 - 81.
     $$
     
   - **Step 4:** Substitute back into the equation:
     
     $$
     16(x + 4)^2 - 256 + 9(y - 3)^2 - 81 = -193.
     $$
     
   - **Step 5:** Combine constant terms:
     
     $$
     -256 - 81 = -337,
     $$
     
     so the equation becomes:
     
     $$
     16(x + 4)^2 + 9(y - 3)^2 - 337 = -193.
     $$
     
   - **Step 6:** Add 337 to both sides:
     
     $$
     16(x + 4)^2 + 9(y - 3)^2 = 144.
     $$
     
   - **Step 7:** Divide both sides by 144:
     
     $$
     \frac{16(x + 4)^2}{144} + \frac{9(y - 3)^2}{144} = 1.
     $$
     
     Simplify the fractions:
     
     $$
     \frac{(x + 4)^2}{9} + \frac{(y - 3)^2}{16} = 1.
     $$
     
   - **Standard Form:** 
     
     $$
     \frac{(x + 4)^2}{9} + \frac{(y - 3)^2}{16} = 1.
     $$
     
   - **Conic Identification:**  
     This is the standard form of an **ellipse**.

1. **Equation:** $$ 16x^2 + 4y^2 = 64 $$ - **Standard Form:** $$ \frac{x^2}{4} + \frac{y^2}{16} = 1 $$ - **Conic:** Ellipse 2. **Equation:** $$ x^2 + y^2 - 6x - 2y = -1 $$ - **Standard Form:** $$ (x - 3)^2 + (y - 1)^2 = 9 $$ - **Conic:** Circle 3. **Equation:** $$ 25x^2 + 9y^2 - 54y = 144 $$ - **Standard Form:** $$ \frac{x^2}{9} + \frac{(y - 3)^2}{25} = 1 $$ - **Conic:** Ellipse 4. **Equation:** $$ 16x^2 + 9y^2 + 128x - 54y = -193 $$ - **Standard Form:** $$ \frac{(x + 4)^2}{9} + \frac{(y - 3)^2}{16} = 1 $$ - **Conic:** Ellipse

Write each equation in standard form. Then identify the
conic as an ellipse or a circle.
11.
12.
13.
14.

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Write each equation in standard form. Then identify the conic as an ellipse or a circle. 11. 12. 13. 14.

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Write each equation in standard form. Then identify the
conic as an ellipse or a circle.
11.
12.
13.
14.