To solve the problem of finding the length of the major axis of the given ellipse, let's break it down step by step:

Given Information:

Focus of the ellipse: $ S = (-1, -1) $

Equation of the directrix: $ x + y + 1 = 0 $

Eccentricity of the ellipse: $ e = \frac{1}{\sqrt{2}} $

Equation of the Ellipse:

The equation connecting the focus, directrix, and eccentricity is:

$ PS = e \cdot PM $

where $ PS $ is the distance from a point $ P(x, y) $ on the ellipse to the focus $ S $, and $ PM $ is the perpendicular distance from $ P $ to the directrix.

Distance $ PS $:

$ PS = \sqrt{(x + 1)^2 + (y + 1)^2} $

Perpendicular distance $ PM $ from $ P $ to the directrix:

$ PM = \frac{|x + y + 1|}{\sqrt{2}} $

Substitute into ellipse equation:

$ \sqrt{(x + 1)^2 + (y + 1)^2} = \frac{1}{\sqrt{2}} \cdot \frac{|x + y + 1|}{\sqrt{2}} $

This simplifies to:

$ \sqrt{(x + 1)^2 + (y + 1)^2} = \frac{x + y + 1}{2} $

Form an equation by solving and simplifying:

Upon squaring both sides and simplifying, the equation becomes:

$ 3x^2 + 3y^2 - 2xy + 6x + 6y + 7 = 0 $

Finding the Center:

To determine the center of the ellipse, we take partial derivatives and solve:

Derivative with respect to $ x $:

$ \frac{\partial f}{\partial x} = 6x - 2y + 6 = 0 $

Derivative with respect to $ y $:

$ \frac{\partial f}{\partial y} = 6y - 2x + 6 = 0 $

Solving these equations simultaneously gives the center $ O \left( -\frac{3}{2}, -\frac{3}{2} \right) $.

Determine the Length of Major Axis:

Distance $ OS $ (from center to the focus):

$ OS = ae = \frac{1}{\sqrt{2}} $

Given:

$ a \cdot \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} $

This implies:

$ a = 1 $

Length of the major axis:

$ 2a = 2 \times 1 = 2 $

Hence, the length of the major axis of the ellipse is 2.

Ellipse

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