y−y 
0
​
 =− 
y 
0
​
 a 
2
 
x 
0
​
 b 
2
 
​
 (x−x 
0
​
 )
Rearranging and simplifying, the equation of the tangent line at 
(
𝑥
0
,
𝑦
0
)
(x 
0
​
 ,y 
0
​
 ) becomes:

𝑥
0
𝑎
2
𝑥
+
𝑦
0
𝑏
2
𝑦
=
1
a 
2
 
x 
0
​
 
​
 x+ 
b 
2
 
y 
0
​
 
​
 y=1

show me the steps

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.

---

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.

Rearranging and simplifying, the equation of the tangent line at (x0, y0) becomes: x0/a^2 * x + y0/b^2 * y = 1. Show me the steps.

Step-by-step solution for deriving the equation of the tangent line to an ellipse at a given point (x0, y0). This solution involves concepts of calculus, implicit differentiation, and conic sections. Suitable for advanced high school or college-level students.

Math Problem Statement

𝑥 0 𝑎 2 𝑥 + 𝑦 0 𝑏 2 𝑦

Solution

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