**Find the Equation of the Ellipse with the Given Information :**

Here we are going to see some practice questions on find the equation of the ellipse with the given information.

**Question 1 :**

Find the equation of the ellipse in each of the cases given below:

(i) foci (± 3 0), e = 1/2

**Solution :**

F_{1} (3, 0) and F_{2} (-3, 0) and e = 1/2

From the given information, we know that the given ellipse is symmetric about x axis.

Midpoint of foci = Center of the ellipse

Center = (3 + (-3))/2, (0 + 0)/2 = C (0, 0)

Distance between foci = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

= √(3 + 3)^{2} + (0 - 0)^{2}

= √6^{2} + (0 - 0)^{2}

2ae = 6

ae = 3

a(1/2) = 3

a = 6

b^{2} = a^{2} (1 - e^{2})

b^{2} = 6^{2} (1 - (1/2)^{2})

b^{2} = 36 (3/4)

b^{2} = 27

(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1

(x^{2}/36) + (y^{2}/27) = 1

(ii) foci (0, ± 4) and end points of major axis are (0, ± 5).

**Solution :**

F_{1} (0, 4) and F_{2} (0, -4)

From the given foci, we know that the ellipse is symmetric about y-axis.

Distance between foci = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

2ae = √(0 - 0)^{2} + (4 + 4)^{2}

2ae = 8

ae = 4

Distance between end points of major axis

= √(0 - 0)^{2} + (5 + 5)^{2}

2a = 10

a = 5

5e = 4

e = 4/5

b^{2 }= a^{2}(1 - e^{2})

b^{2 }= 5^{2} (1 - (4/5)^{2})

b^{2 }= 5^{2} (9/25)

b^{2 }= 9

Hence the required equation of ellipse is

(x^{2}/25) + (y^{2}/9) = 1

(iii) length of latus rectum 8, eccentricity = 3/5 and major axis on x -axis.

**Solution :**

Length of latus rectum = 8

2b^{2}/a = 8

b^{2} = 4a

e = 3/5

Since the major axis is on x-axis, the ellipse is symmetric about x-axis.

b^{2} = a^{2} (1 - e^{2})

4a = a^{2} (1 - (3/5)^{2})

4 = a (16/25)

a = 25/4

b^{2} = 4(25/4)

b^{2} = 25

(x^{2}/(16/625)) + (y^{2}/25) = 1

(625x^{2}/16) + (y^{2}/25) = 1

(iv) length of latus rectum 4 , distance between foci 4√2 and major axis as y - axis.

**Solution :**

length of latus rectum = 4

2b^{2}/a = 4

b^{2} = 2a -------(1)

Distance between foci = 4√2

2ae = 4√2

ae = 2√2

b^{2 } = a^{2} (1 - e^{2})

b^{2 } = a^{2} - (ae)^{2}

b^{2 } = a^{2} - (2√2)^{2}

b^{2 } = a^{2} - 8 -------(2)

2a = a^{2} - 8

a^{2 }- 2a - 8 = 0

(a - 4) (a + 2) = 0

a = 4 and a = -2

If a = 4, then b^{2} = 2(4) = 8

If a = -2, then b^{2} = 2(-2) = -4 (Not admissible)

(x^{2}/8) + (y^{2}/16) = 1

After having gone through the stuff given above, we hope that the students would have understood, "Find the Equation of the Ellipse with the Given Information".

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