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(4) 4. Find the -value(s) where has vertical tangents.

Ask by Paul Rose. in the United States

Jan 22,2025

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Free Response With Calculator (4) 4. Find the $$ y $$-value(s) where $$ x^{2}+y^{2}=-4 x-6 y-4 $$ has vertical tangents.

UpStudy AI · Accepted Answer

To find the $$ y $$-values where the equation $$ x^{2} + y^{2} = -4x - 6y - 4 $$ has vertical tangents, follow these steps:

1. **Rewrite the Equation**:
   $$
   x^{2} + 4x + y^{2} + 6y + 4 = 0
   $$
   
2. **Complete the Square**:
   - For $$ x $$:
     $$
     x^{2} + 4x = (x + 2)^2 - 4
     $$
   - For $$ y $$:
     $$
     y^{2} + 6y = (y + 3)^2 - 9
     $$
   
3. **Substitute Back**:
   $$
   (x + 2)^2 - 4 + (y + 3)^2 - 9 + 4 = 0 \
   \Rightarrow (x + 2)^2 + (y + 3)^2 = 9
   $$
   
   This represents a circle with center $$(-2, -3)$$ and radius $$3$$.

4. **Determine Vertical Tangents**:
   Vertical tangents occur where the derivative $$ \frac{dy}{dx} $$ is undefined, which corresponds to the leftmost and rightmost points of the circle.
   
   - **Leftmost Point**:
     $$
     x = -2 - 3 = -5 \
     \Rightarrow (y + 3)^2 = 0 \
     y = -3
     $$
   
   - **Rightmost Point**:
     $$
     x = -2 + 3 = 1 \
     \Rightarrow (y + 3)^2 = 0 \
     y = -3
     $$
   
   Both points yield $$ y = -3 $$.

**Answer**:  
The $$ y $$-value where there are vertical tangents is $$ \boxed{-3} $$.
The $$ y $$-value where there are vertical tangents is $$ y = -3 $$.

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(4) 4. Find the -value(s) where has vertical tangents.

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Free Response With Calculator (4) 4. Find the -value(s) where has vertical tangents.