Name: Illustrations on the Standard Equation of Ellipse
Uploaded: 2019-07-19
Duration: 3 min 22 s
Description: Let's try to find parts of the ellipse, such as focus, directrix, eccentricity & length of Latus Rectum through an illustration. Watch this video and learn.

Question

Find the equation of the ellipse in the following cases:
focus is

(- 2, 3),

directrix is

2 x + 3 y + 4 = 0

and

e = \frac{4}{5}

Accepted Answer

Let $S (- 2, 3)$ be the focus and $ZZ$ ' be the directrix.
Question Let $P (x, y)$ be any point on the ellipse and let $PM$ be the perpendicular from $P$ on the directrix. Then by the definition, we have: $SP = e \times PM$

$\Rightarrow SP = \frac{4}{5} \times PM$

$\Rightarrow \frac{5}{4} S P = PM$

$\Rightarrow \frac{25}{16} (SP)^{2} = {PM}^{2}$

$\Rightarrow \frac{25}{16} [(x + 2)^{2} + (y - 3)^{2}] = {(\frac{|2 x + 3 y + 4|}{\sqrt{2^{2} + 3^{2}}})}^{2}$

$\Rightarrow \frac{25}{16} [x^{2} + 4 + 4 x + y^{2} + 9 - 6 y] = \frac{4 x^{2} + 9 y^{2} + 16 + 12 x y + 24 y + 16 x}{13}$

$\Rightarrow 325 (x^{2} + 4 + 4 x + y^{2} + 9 - 6 y) = 16 (4 x^{2} + 9 y^{2} + 16 + 12 x y + 24 y + 16 x)$

$\Rightarrow 325 x^{2} + 1300 + 1300 x + 325 y^{2} + 2925 - 1950 y = 64 x^{2} + 144 y^{2} + 256 + 192 x y + 384 y + 256 x$

$\Rightarrow 261 x^{2} + 181 y^{2} + 1044 x - 2309 y - 192 x y + 3969 = 0$

This is the required equation of the ellipse.

Find the equation of the ellipse in the following cases:
focus is $(- 2, 3),$ directrix is $2 x + 3 y + 4 = 0$ and $e = \frac{4}{5}$

Important Questions on Ellipse

Learn and Prepare for any exam you want !

Find the equation of the ellipse in the following cases: focus is (-2,3), directrix is 2x+3y+4=0 and e=45

Important Questions on Ellipse

Learn and Prepare for any exam you want !

Find the equation of the ellipse in the following cases:
focus is $(- 2, 3),$ directrix is $2 x + 3 y + 4 = 0$ and $e = \frac{4}{5}$