Question

The equation of an ellipse which has a focus (6, 7), a directix x + y + 2 = 0 and eccentricity \(\frac{1}{{\sqrt 3 }}\), is:
1. 5x² - 2xy + 5y² - 76x - 88y + 506 = 0
2. 5x2 - 2xy + 5y2 - 76x - 88y + 204 = 0
3. 5x2 - 2xy + 5y2 - 76x + 88y + 204 = 0
4. 5x2 - 2xy + 5y2 = 0

Answer 1

Correct Answer - Option 1 : 5x² - 2xy + 5y² - 76x - 88y + 506 = 0

Concept:

Standard equation of ellipse: 

\(\rm \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

Eccentricity,

e = \(\rm \sqrt{1-\frac{a^2}{b^2}}\), for a < b

If a < b, focus = (0, ±ve)

Eccentricity: the ratio of the distance between a point on the ellipse and its focus to the distance between the point and the directrix is called its eccentricity.

Calculation:

Given: focus (x1, y1) = (6, 7), a directix x + y + 2 = 0 and eccentricity \(\frac{1}{{\sqrt 3 }}\) then,

\(⇒ \frac{\sqrt {(x-6)^2 + (y - 7)^2 }}{\frac{(x+y+2)}{\sqrt{2}}}=\frac{1}{\sqrt3}\)

\(⇒ \sqrt 2 \sqrt {\frac{(x-6)^2 + (y-7)^2}{(x+y+2)2}}=\frac{1}{\sqrt3}\)

on squaring both the side,

⇒ (x - 6)2 + (y - 7)2 = \(\frac{1}{6}\)(x + y + z)2

⇒ (x2 - 12x + 36) + (y2 - 14y + 49) = \(\frac{1}{6}\)(x2 + y2 + 4 + 2xy + 4x + 4y)

⇒ 6(x2 - 12x + 36 + y2 - 14y + 49) = (x2 + y2 + 4 + 2xy + 4x + 4y)

⇒ 6x2 - 72x + 216 + 6y2 - 84y + 294 = x2 + y2 + y + 2xy + 4x + 4y

⇒ 5x2 + - 2xy + 5y2 - 76x - 88y + 506 = 0