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To find all points on the given curve \( x^2 - 3xy + 4y^2 = 7 \) that have vertical tangent lines, we need to determine where the derivative \(\frac{dx}{dy}\) (the slope of the tangent line) is undefined, as vertical tangent lines correspond to an undefined derivative in terms of \( \frac{dy}{dx} \).

### Steps to Solve:

1. **Implicit Differentiation**:
   Differentiate both sides of the equation \( x^2 - 3xy + 4y^2 = 7 \) implicitly with respect to \(x\).

   \[
   2x - 3(x \frac{dy}{dx} + y) + 8y \frac{dy}{dx} = 0
   \]

2. **Solve for \( \frac{dy}{dx} \)**:
   Rearrange the above expression to isolate \( \frac{dy}{dx} \) on one side.

   \[
   2x - 3y + (-3x + 8y) \frac{dy}{dx} = 0
   \]

   Now, solve for \( \frac{dy}{dx} \):

   \[
   (-3x + 8y) \frac{dy}{dx} = 3y - 2x
   \]

   \[
   \frac{dy}{dx} = \frac{3y - 2x}{-3x + 8y}
   \]

3. **Find Vertical Tangent Points**:
   A vertical tangent occurs when \( \frac{dy}{dx} \) is undefined, which happens when the denominator equals zero. Set the denominator \( -3x + 8y = 0 \):

   \[
   -3x + 8y = 0
   \]

   Solving for \( x \):

   \[
   x = \frac{8}{3} y
   \]

4. **Substitute into the Original Equation**:
   Substitute \( x = \frac{8}{3} y \) into the original equation \( x^2 - 3xy + 4y^2 = 7 \):

   \[
   \left( \frac{8}{3}y \right)^2 - 3\left(\frac{8}{3}y\right)y + 4y^2 = 7
   \]

   Simplifying each term:

   \[
   \frac{64}{9} y^2 - 8y^2 + 4y^2 = 7
   \]

   Combine the terms:

   \[
   \frac{64}{9} y^2 - \frac{72}{9} y^2 + \frac{36}{9} y^2 = 7
   \]

   \[
   \frac{28}{9} y^2 = 7
   \]

   Solve for \( y^2 \):

   \[
   y^2 = \frac{7 \times 9}{28} = \frac{63}{28} = \frac{9}{4}
   \]

   So,

   \[
   y = \pm \frac{3}{2}
   \]

5. **Find the Corresponding \( x \)-Values**:
   Now substitute \( y = \pm \frac{3}{2} \) back into \( x = \frac{8}{3}y \):

   - For \( y = \frac{3}{2} \), \( x = \frac{8}{3} \times \frac{3}{2} = 4 \).
   - For \( y = -\frac{3}{2} \), \( x = \frac{8}{3} \times -\frac{3}{2} = -4 \).

### Conclusion:
The points where the curve has vertical tangent lines are \( (4, \frac{3}{2}) \) and \( (-4, -\frac{3}{2}) \).

Would you like a more detailed explanation on any of these steps or further clarifications?

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Here are five related questions to deepen your understanding:
1. How do you find horizontal tangents for the same curve?
2. What is the geometric significance of vertical tangents in implicit curves?
3. How would the analysis change if the curve equation were slightly altered?
4. Can you find points where the tangent line is horizontal instead of vertical?
5. What does the discriminant of an implicit curve tell us about its shape?

**Tip:** When solving problems involving implicit differentiation, always check for cases where the derivative becomes undefined to find vertical tangents.

Find all points on x^2 - 3xy + 4y^2 = 7 that have vertical tangent lines.

Learn how to find points where the curve x^2 - 3xy + 4y^2 = 7 has vertical tangent lines using implicit differentiation. This step-by-step solution covers key mathematical concepts including implicit differentiation, algebraic manipulation, and substitution. Suitable for Grade 12 and college-level students.

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