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To find the tangent line to the ellipse defined by \( 3x^2 + 5y^2 = 17 \) at the point \((-2, 1)\), we can proceed by using implicit differentiation.

### Step 1: Differentiate the Equation Implicitly
Given:
\[
3x^2 + 5y^2 = 17
\]
we differentiate both sides with respect to \(x\):

\[
\frac{d}{dx}(3x^2) + \frac{d}{dx}(5y^2) = \frac{d}{dx}(17)
\]

This yields:
\[
6x + 10y \frac{dy}{dx} = 0
\]

### Step 2: Solve for \(\frac{dy}{dx}\)
Rearrange the equation to isolate \(\frac{dy}{dx}\):
\[
10y \frac{dy}{dx} = -6x
\]
\[
\frac{dy}{dx} = -\frac{6x}{10y} = -\frac{3x}{5y}
\]

### Step 3: Substitute the Point \((-2, 1)\) to Find the Slope
Now we substitute \(x = -2\) and \(y = 1\) into the derivative to find the slope of the tangent line at this point:

\[
\frac{dy}{dx} = -\frac{3(-2)}{5(1)} = \frac{6}{5}
\]

### Step 4: Write the Equation of the Tangent Line
The slope of the tangent line at \((-2, 1)\) is \(\frac{6}{5}\). Using the point-slope form of the line equation, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (-2, 1)\) and \(m = \frac{6}{5}\), we get:

\[
y - 1 = \frac{6}{5}(x + 2)
\]

Expanding and simplifying:
\[
y - 1 = \frac{6}{5}x + \frac{12}{5}
\]
\[
y = \frac{6}{5}x + \frac{12}{5} + 1
\]
\[
y = \frac{6}{5}x + \frac{17}{5}
\]

### Final Answer
The equation of the tangent line to the ellipse at the point \((-2, 1)\) is:
\[
y = \frac{6}{5}x + \frac{17}{5}
\]

Would you like further details or have any questions?

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Here are some related questions you might explore:
1. How do you find the normal line to the ellipse at a given point?
2. How would the process change if the ellipse were centered at a different point?
3. How can you verify that a given point lies on an ellipse?
4. What are the major and minor axes of an ellipse?
5. How does implicit differentiation work for functions defined by equations?

**Tip:** Implicit differentiation is especially useful for curves where \(y\) is not explicitly defined as a function of \(x\), such as ellipses and circles.

Find the tangent line to the ellipse defined by 3x^2 + 5y^2 = 17 at the point (-2, 1).

Learn how to find the tangent line to the ellipse 3x^2 + 5y^2 = 17 at the point (-2, 1) using implicit differentiation. This solution covers key calculus concepts, including implicit differentiation and the point-slope form of a line, suitable for advanced high school students in Grade 12.

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