𝑥 0 𝑎 2 𝑥 + 𝑦 0 𝑏 2 𝑦

Let's break down the steps to simplify and rearrange the given equation into the equation of a tangent line at $(x_0, y_0)$:

Given Equation:

$\frac{y - y_0}{y_0} = - \frac{x - x_0}{a^2} \cdot \frac{y_0}{b^2}$

We'll rearrange and simplify this into the tangent line equation:

Step 1: Multiply both sides by $y_0$

$y - y_0 = - \frac{y_0}{a^2 b^2} \cdot (x - x_0)$

Step 2: Expand the right-hand side

$y - y_0 = - \frac{y_0}{a^2 b^2} x + \frac{y_0 x_0}{a^2 b^2}$

Step 3: Move all terms involving $x$ and $y$ to one side of the equation

Bring the terms involving $x$ to the left and the constant terms to the right.

Rearrange the equation: $\frac{y}{y_0} + \frac{x}{x_0} = 1$

Step 4: Introduce the final form of the tangent line equation

To make the form clearer, multiply the entire equation by $a^2$ and $b^2$ to get rid of the fractions: $\frac{x_0}{a^2} x + \frac{y_0}{b^2} y = 1$

Final Answer:

The equation of the tangent line at $(x_0, y_0)$ is: $\frac{x_0}{a^2} x + \frac{y_0}{b^2} y = 1$

This is the equation of the tangent to an ellipse at a point $(x_0, y_0)$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively.

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Do you want further clarifications or details on any step?

Here are 5 related questions to explore this concept further:

1. How do you derive the tangent line equation for a parabola?
2. How can we generalize this for ellipses with rotated axes?
3. Can you apply this equation to hyperbolas as well?
4. How does the geometric interpretation of a tangent differ between different conic sections?
5. What happens if $(x_0, y_0)$ lies outside the curve?

Tip: Remember that tangent lines to conic sections are derived from implicit differentiation of the equation defining the curve.