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only give me the solutions without the explanations

Here are the solutions to the given problems:

### (a) If \( x^2 + y^2 = 400 \), find \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = -\frac{x}{y}
\]

### (b) Find the equation of the tangent to the circle \( x^2 + y^2 = 400 \) at the point (16, 12):

At \( (x, y) = (16, 12) \):

\[
\frac{dy}{dx} = -\frac{16}{12} = -\frac{4}{3}
\]

The equation of the tangent line is:

\[
y - 12 = -\frac{4}{3}(x - 16)
\]

or

\[
4x + 3y = 100
\]

Let me know if you'd like further details or have any questions!

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Here are 5 related questions:
1. How do you apply implicit differentiation to solve for \( \frac{dy}{dx} \)?
2. How can you find the tangent line to any circle at a given point?
3. What is the geometric significance of \( \frac{dy}{dx} \) for a circle equation?
4. How do you solve for the slope of a tangent line using derivatives?
5. Can you derive the general formula for the tangent to a circle?

**Tip:** Implicit differentiation is useful for equations where \( y \) is not explicitly written as a function of \( x \), such as circles or ellipses.

If x^2 + y^2 = 400, find dy/dx. Find the equation of the tangent to the circle x^2 + y^2 = 400 at the point (16, 12).

Learn how to find the derivative using implicit differentiation and calculate the equation of a tangent line to the circle x^2 + y^2 = 400 at the point (16, 12). This problem covers the concepts of implicit differentiation, tangent lines, and geometry of circles. Suitable for Grades 10-12.

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