Question

One of the foci of an ellipse is \( (2,-3) \) and its corresponding directrix is \( 2x+y=5 \). If the eccentricity of the ellipse is \( \frac{\sqrt{5}}{3} \) then the coordinates of the other focus are

• A. \( (18, 5) \)
• B. \( (4, -2) \)
• C. \( (-2, -5) \)
• D. \( (-4, -6) \)

Hint:

Using the relation \( 9\vec{S} = 4\vec{C} + 5\vec{Z} \) (derived from section formula) is more efficient than distance equations involving variables.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:

We find the center of the ellipse first. The center lies on the major axis (line passing through the focus and perpendicular to the directrix). We can determine the center's location using the relationship between the focus, center, and directrix distances. Once the center is found, the other focus can be found using the midpoint formula.

Step 2: Key Formula or Approach:

1. Distance from center to focus = ae.
2. Distance from center to directrix = a/e.
3. Section formula or vector approach to find the center.

Step 3: Detailed Explanation:

Let S(2, -3) be the focus and the directrix be 2x + y - 5 = 0.

Let Z be the foot of the perpendicular from S to the directrix.

Distance SZ (perpendicular distance from S to line):
SZ = |2(2) + (-3) - 5| / √(2² + 1²) = 4 / √5

On the major axis:
CS = ae
CZ = a/e

So,
SZ = CZ - CS = a(1/e - e)

Substitute e = √5 / 3:
4/√5 = a(3/√5 - √5/3)
= a[(9 - 5) / (3√5)]
= 4a / (3√5)

4/√5 = 4a / (3√5)
a = 3

Now find distances from center:
CS = ae = 3(√5/3) = √5
CZ = a/e = 3(3/√5) = 9/√5

Since S lies between C and Z,
ratio CS : SZ = √5 : (4/√5) = 5 : 4

Using section/vector form:
9S = 4C + 5Z
⇒ 4C = 9S - 5Z

Find coordinates of Z:

Axis line (perpendicular to 2x + y = 5 through (2, -3)):
x - 2y = 8

Solve:
2x + y = 5
x - 2y = 8

Multiply first by 2:
4x + 2y = 10
Add:
5x = 18 ⇒ x = 3.6
y = 5 - 2(3.6) = -2.2

So,
Z(3.6, -2.2)

Find Center C:

4xC = 9(2) - 5(3.6) = 18 - 18 = 0 ⇒ xC = 0
4yC = 9(-3) - 5(-2.2) = -27 + 11 = -16 ⇒ yC = -4

So,
C(0, -4)

Find other focus S':

Center is midpoint of S and S':
(2 + x') / 2 = 0 ⇒ x' = -2
(-3 + y') / 2 = -4 ⇒ y' = -5

So,
S'(-2, -5)

Step 4: Final Answer:

The coordinates are (-2, -5).