An equation of the normal to the ellipse x²/a² + y²/b² = 1 with eccentricity e at the positive end of the latus rectum is

• A: x + ey + e³a = 0
• B: x - ey - e³a = 0
• C: x - ey + e³a = 0
• D: x + ey - e³a = 0

To find the equation of the normal to the ellipse given by the equation x²/a² + y²/b² = 1 at the positive end of the latus rectum, we first need to identify the coordinates of that point. The latus rectum of the ellipse is located at the points (ae, b²/a) and (-ae, b²/a), where e is the eccentricity.

Finding the Normal Equation

At the positive end of the latus rectum, the coordinates are (ae, b²/a). The slope of the tangent line at this point can be derived from the derivative of the ellipse equation. The slope of the normal line is the negative reciprocal of the tangent slope.

Normal Line Equation

The general form of the normal line at a point (x₀, y₀) is given by:

• y - y₀ = m(x - x₀), where m is the slope of the normal.

Substituting the coordinates and the slope, we can rearrange the equation into the standard form.

Identifying the Correct Option

After performing the calculations, the equation of the normal line simplifies to:

• x + ey + e³a = 0

Thus, the correct answer is:

Final Answer

A: x + ey + e³a = 0