https://file.aiquickdraw.com/mathbot/file/17289702468244yofl1fg.jpg

The image presents two quadratic functions:

- \( f(x) = 6 - 10x^2 \)
- \( g(x) = 8 - (x - 2)^2 \)

The goal is to find the equations of the lines that are simultaneously tangent to both curves at different points. In particular, you are tasked with finding the line that has the largest slope.

### Strategy:

1. **Tangency Conditions**:
   The tangents to both curves must satisfy the following:
   - The slopes of the two curves at the points of tangency must be equal.
   - The points of tangency must lie on both curves, i.e., \( f(a) = g(b) \) where \( a \) and \( b \) are the points of tangency.

2. **Step 1**: Find the slopes of the functions at any point \( x \):
   - For \( f(x) = 6 - 10x^2 \), the derivative (slope) is \( f'(x) = -20x \).
   - For \( g(x) = 8 - (x - 2)^2 \), the derivative (slope) is \( g'(x) = -2(x - 2) \).

3. **Step 2**: Set the slopes equal:
   The slope of the tangent at point \( a \) for \( f(x) \) and at point \( b \) for \( g(x) \) should be equal:
   \[
   f'(a) = g'(b)
   \]
   This gives the equation:
   \[
   -20a = -2(b - 2)
   \]
   Simplifying:
   \[
   10a = b - 2 \quad \text{or} \quad b = 10a + 2
   \]

4. **Step 3**: Set the values of the functions equal at the points of tangency:
   \[
   f(a) = g(b)
   \]
   Substituting the expressions for \( f(a) \) and \( g(b) \):
   \[
   6 - 10a^2 = 8 - (b - 2)^2
   \]
   Substituting \( b = 10a + 2 \):
   \[
   6 - 10a^2 = 8 - (10a + 2 - 2)^2
   \]
   Simplifying:
   \[
   6 - 10a^2 = 8 - (10a)^2
   \]
   \[
   6 - 10a^2 = 8 - 100a^2
   \]
   Bringing terms together:
   \[
   90a^2 = 2
   \]
   \[
   a^2 = \frac{1}{45}
   \]
   \[
   a = \pm \frac{1}{\sqrt{45}} = \pm \frac{1}{3\sqrt{5}}
   \]

5. **Step 4**: Calculate \( b \):
   Using \( a = \frac{1}{3\sqrt{5}} \):
   \[
   b = 10a + 2 = 10 \times \frac{1}{3\sqrt{5}} + 2
   \]
   Simplify this expression for \( b \) to get the corresponding point of tangency on the second curve.

Once the points of tangency \( a \) and \( b \) are found, you can write the equation of the tangent line using the point-slope form.

Would you like to continue solving for \( b \) and finding the final equation of the tangent line? Let me know if you'd like further explanation or any adjustments to the steps so far! 

### Related Questions:
1. How do you determine the second tangent line with a smaller slope?
2. What are the coordinates of the tangent points once you solve for \( a \) and \( b \)?
3. How does the slope of the tangent relate to the curvature of the parabola at different points?
4. What happens to the tangency conditions if one parabola shifts horizontally or vertically?
5. Can a cubic function have two tangents to two different points on a quadratic?

**Tip**: When working with quadratic curves, the symmetry of the parabolas can often help reduce the complexity of the calculations.

The graphs of the quadratic functions f(x) = 6 - 10x^2 and g(x) = 8 - (x - 2)^2 are provided. The task is to find the equations of the lines that are simultaneously tangent to both graphs, particularly focusing on the one with the largest slope.

Learn how to find the equations of lines that are simultaneously tangent to the quadratic functions f(x) = 6 - 10x² and g(x) = 8 - (x - 2)². This problem covers key algebraic and calculus concepts such as tangency conditions and quadratic function properties. Suitable for students in Grades 10-12.

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