Question

The equation of ellipse with focus (- 1, 1), directrix x – y + 3 = 0 and eccentricity 1/2 is

A. 7x² + 2xy + 7y² + 10x + 10y + 7 = 0

B. 7x² + 2xy + 7y² + 10x - 10y + 7 = 0

C. 7x² + 2xy + 7y² + 10x - 10y - 7 = 0

D. None of these

Answer 1

Correct option is B. 7x² + 2xy + 7y² + 10x - 10y + 7 = 0

Given that we need to find the equation of the ellipse whose focus is S(- 1,1) and directrix(M) is x - y + 3 = 0 and eccentricity(e) is equal to 1/2.

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

We know that the distance between the focus and any point on ellipse is equal to the eccentricity times the perpendicular distance from that point to the directrix.

We know that distance between the points (x₁, y₁) and (x₂, y₂) is \(\sqrt{(\text x_1-\text x_2)^2+(y_1-y_2)^2}.\)

We know that the perpendicular distance from the point (x₁,y₁) to the line ax + by + c = 0 is \(\cfrac{|a\text x_1+by_1+c|}{\sqrt{a^2+b^2}}\)

⇒ SP = ePM

⇒ SP² = e²PM²

⇒ 8x² + 8y² + 16x - 16y + 16

= x² + y² - 2xy - 6y + 6x + 9

⇒ 7x² + 7y² + 2xy + 10x - 10y + 7 = 0