Question

A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 12. Then the length of the semi-major axis is

Answer 1

Answer:

8/3

Step-by-step explanation:

Hi,

1st Method

From the definition of conic section, if S is the focus of the

ellipse and P is any

general point on the ellipse and line L is the directrix,

then SP/PZ = e, where e is the eccentricity of the ellipse and PZ

is the

perpendicular distance from P to the line L(directrix) .

=>SP² = e²PZ²

Given e = 1/2

L: x - 4 = 0

PZ = |x - 4|

SP² = (x - 0)² + (y - 0)² = x² + y²,

Thus,

x² + y² = 1/4*|x - 4|²

⇒ 4x² + 4y² = x² - 8x + 16

⇒ 3x² + 8x + 4y² = 16

⇒3(x² + 8x/3) + 4y² = 16

⇒3(x + 4/3)² + 4y² = 16 + 16/3 = 64/3

⇒(x + 4/3)²/(8/3)² + y²/(4/√3)² = 1

Hence , the above equation is in the standard form,

where a= semi-major axis = 8/3.

2nd Method

Distance from focus to the corresponding directrix = a(1/e - e)

Now, given e = 1/2 and distance from focus (0,0) to the directrix

is 4

hence a(2 - 1/2) = 4

⇒3a/2 = 4

⇒ a = 8/3

Hope, it helped.