Finding the Equation of Ellipse from Focus, Directrix, and Eccentricity

To find the equation of an ellipse given its focus, directrix, and eccentricity, we use the fundamental definition of a conic section: the locus of a point P such that its distance from a fixed point (the focus) is a constant multiple (the eccentricity) of its distance from a fixed line (the directrix).

Let P(x, y) be any point on the ellipse.

The given focus is S = (6, 7).

The given directrix is the line with equation x + y + 2 = 0.

The given eccentricity is \(e = \frac{1}{{\sqrt 3 }}\).

Applying the Conic Section Definition

The definition states that the distance from P to S (PS) is equal to the eccentricity \(e\) times the distance from P to the directrix (PM), where PM is the perpendicular distance from P to the directrix.

So, we have the relation:

\(PS = e \cdot PM\)

Calculating PS (Distance from P to Focus)

Using the distance formula for points P(x, y) and S(6, 7):

\(PS = \sqrt{(x - 6)^2 + (y - 7)^2}\)

Calculating PM (Distance from P to Directrix)

The directrix is the line \(x + y + 2 = 0\). The perpendicular distance from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by \(\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}\).

For P(x, y) and the line \(x + y + 2 = 0\) (where A=1, B=1, C=2):

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

Setting up the Equation of Ellipse

Substitute the expressions for PS, PM, and the value of \(e\) into the equation \(PS = e \cdot PM\):

\(\sqrt{(x - 6)^2 + (y - 7)^2} = \frac{1}{{\sqrt 3 }} \cdot \frac{|x + y + 2|}{\sqrt{2}}\)

Squaring Both Sides

To remove the square root, square both sides of the equation:

\(\left(\sqrt{(x - 6)^2 + (y - 7)^2}\right)^2 = \left(\frac{1}{{\sqrt 3 }} \cdot \frac{|x + y + 2|}{\sqrt{2}}\right)^2\)

\((x - 6)^2 + (y - 7)^2 = \left(\frac{1}{{\sqrt 3 }}\right)^2 \cdot \left(\frac{x + y + 2}{\sqrt{2}}\right)^2\)

\((x^2 - 12x + 36) + (y^2 - 14y + 49) = \frac{1}{3} \cdot \frac{(x + y + 2)^2}{2}\)

\(x^2 - 12x + 36 + y^2 - 14y + 49 = \frac{1}{6} (x^2 + y^2 + 2^2 + 2(x)(y) + 2(x)(2) + 2(y)(2))\)

\(x^2 + y^2 - 12x - 14y + 85 = \frac{1}{6} (x^2 + y^2 + 4 + 2xy + 4x + 4y)\)

\(x^2 + y^2 - 12x - 14y + 85 = \frac{1}{6} (x^2 + y^2 + 2xy + 4x + 4y + 4)\)

Simplifying to Find the Equation of Ellipse

Multiply both sides by 6 to clear the fraction:

\(6(x^2 + y^2 - 12x - 14y + 85) = x^2 + y^2 + 2xy + 4x + 4y + 4\)

\(6x^2 + 6y^2 - 72x - 84y + 510 = x^2 + y^2 + 2xy + 4x + 4y + 4\)

Move all terms to the left side of the equation:

\(6x^2 - x^2 + 6y^2 - y^2 - 2xy - 72x - 4x - 84y - 4y + 510 - 4 = 0\)

\(5x^2 + 5y^2 - 2xy - 76x - 88y + 506 = 0\)

Rearranging the terms to match the standard form of the equation of ellipse:

\(5x^2 - 2xy + 5y^2 - 76x - 88y + 506 = 0\)

This is the required equation of ellipse.