1.0 Definition

  Ellipse

An ellipse is the locus of a point which moves in a plane such that its distance from a fixed point

(i.e., focus) is at a constant ratio from a fixed line (i.e., directrix). The ratio is called eccentricity $(e)$.

$$\frac{{SP}}{{PM}} = e$$

For ellipse, $e < 1$.

Question 1. Find the equation of ellipse whose focus is $(1,1)$, eccentricity $e = \frac{1}{2}$ and equation of directrix is $x - y + 3 = 0$.

Solution: As given in the question coordinates of focus is $S(1,1)$. From the definition of ellipse,

$$\frac{{SP}}{{PM}} = e$$

or,

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

Squaring both sides, we get

$$\begin{equation} \begin{aligned} {x^2} + {y^2} - 2x - 2y + 2 = \frac{1}{4} \times \frac{{{{(x - y + 3)}^2}}}{2} \\ 8{x^2} + 8{y^2} - 16x - 16y + 16 = {x^2} + {y^2} - 2xy + 9 + 6x - 6y \\ 7{x^2} + 7{y^2} + 2xy - 22x - 10y + 7 = 0 \\\end{aligned} \end{equation} $$