Prove that the equation of tangent of the ellipse x²/a² + y²/b² = 1 at point (x₁, y₁) is (x/x₁)² + (y/y₁)² = 1.

To prove that the equation of the tangent to the ellipse given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) at the point \( (x_1, y_1) \) is \( \left(\frac{x}{x_1}\right)^2 + \left(\frac{y}{y_1}\right)^2 = 1 \), we start with the standard form of the ellipse.

Step 1: Identify the Point on the Ellipse

First, ensure that the point \( (x_1, y_1) \) lies on the ellipse. This means it must satisfy the ellipse equation:

• Substituting \( x_1 \) and \( y_1 \) into the ellipse equation gives:
• \( \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1 \)

Step 2: Derive the Tangent Equation

The general equation of the tangent to the ellipse at point \( (x_1, y_1) \) can be derived using the formula:

\( \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \)

Step 3: Rearranging the Tangent Equation

To transform this into the desired form, we can manipulate the equation:

• Multiply both sides by \( a^2b^2 \) to eliminate the denominators:
• \( b^2xx_1 + a^2yy_1 = a^2b^2 \)

Step 4: Normalize the Equation

Now, we can express this in a normalized form. Divide the entire equation by \( a^2b^2 \):

\( \frac{b^2x^2}{a^2b^2} + \frac{a^2y^2}{a^2b^2} = 1 \)

This simplifies to:

\( \left(\frac{x}{x_1}\right)^2 + \left(\frac{y}{y_1}\right)^2 = 1 \)

Final Result

Thus, we have shown that the equation of the tangent to the ellipse at the point \( (x_1, y_1) \) is indeed:

\( \left(\frac{x}{x_1}\right)^2 + \left(\frac{y}{y_1}\right)^2 = 1 \)