**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse [(x**^{2})/_{36}] + [(y^{2})/_{16}]= 1

Solution:

The given equation is [(x^{2})/_{36}] + [(y^{2})/_{16}]= 1

Here the denominator of [(x^{2})/_{36}] is greater than the denominator of [(y^{2})/_{16}]

Therefore, the major axis is along the x- axis, while the minor axis is along the y-axis

On comparing the given equation with [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1

Therefore, c = √(a^{2} – b^{2}) = √(36-16) = 2√5

We get, a = 6 and b = 4

Thus, the coordinates of the foci are (2√5, 0) and (-2√5, 0)

The coordinates of the vertices are (6, 0) and (-6, 0)

Length of major axis = 2a = 12

Length of minor axis = 2b = 8

Eccentricity e = ^{c}/_{a} = ^{2√5}/_{6} = ^{√5}/_{3}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 16}/_{6} = ^{16}/_{3}

**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse [(x**^{2})/_{4}] + [(y^{2})/_{25}]= 1

Solution:

The given equation is [(x^{2})/_{4}] + [(y^{2})/_{25}]= 1

Here the denominator of [(y^{2})/_{25}] is greater than the denominator of [(x^{2})/_{4}]

Therefore, the major axis is along the y- axis, while the minor axis is along the x-axis

On comparing the given equation with [(x^{2})/(b^{2})] + [(y^{2})/ (a^{2})]= 1

We get, a = 5 and b = 2

c = √(a^{2} – b^{2}) = √(25-4) = √21

Thus, the coordinates of the foci are (0, √21) and (0, -√21)

The coordinates of the vertices are (0, 5) and (0, -5)

Length of major axis = 2a = 10

Length of minor axis = 2b = 4

Eccentricity e = ^{c}/_{a} = ^{√21}/_{5}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 4}/_{5} = ^{8}/_{5}

**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse [(x**^{2})/_{16}] + [(y^{2})/_{9}]= 1

Solution:

The given equation is [(x^{2})/_{16}] + [(y^{2})/_{9}]= 1

Here the denominator of [(x^{2})/_{16}] is greater than the denominator of [(y^{2})/_{9}]

Therefore, the major axis is along the x- axis, while the minor axis is along the y-axis

On comparing the given equation with [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1

We get, a = 4 and b = 3

Therefore, c = √(a^{2} – b^{2}) = √(16-9) =√7

Thus, the coordinates of the foci are (√7, 0) and (-√7, 0)

The coordinates of the vertices are (4, 0) and (-4, 0)

Length of major axis = 2a = 8

Length of minor axis = 2b = 6

Eccentricity e = ^{c}/_{a} = ^{√7}/_{4}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 9}/_{4} = ^{9}/_{4}

**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse [(x**^{2})/_{25}] + [(y^{2})/_{100}]= 1

Solution:

The given equation is [(x^{2})/_{25}] + [(y^{2})/_{100}]= 1

Here the denominator of [(y^{2})/_{100}] is greater than the denominator of [(x^{2})/_{25}]

Therefore, the major axis is along the y- axis, while the minor axis is along the x-axis

On comparing the given equation with [(x^{2})/(b^{2})] + [(y^{2})/ (a^{2})]= 1

Therefore, c = √(a^{2} – b^{2}) = √(100-25) = √75 = 5√3

We get, a = 5 and b = 10

Thus, the coordinates of the foci are (0, 5√3) and (0, -5√3)

The coordinates of the vertices are (0, 10) and (0, -10)

Length of major axis = 2a = 20

Length of minor axis = 2b = 10

Eccentricity e = ^{c}/_{a} = ^{5√3}/_{10} = ^{√3}/_{2}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 25}/_{10} =5

**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse [(x**^{2})/_{49}] + [(y^{2})/_{36}]= 1

Solution:

The given equation is [(x^{2})/_{49}] + [(y^{2})/_{36}]= 1

Here the denominator of [(x^{2})/_{49}] is greater than the denominator of [(y^{2})/_{36}]

Therefore, the major axis is along the x- axis, while the minor axis is along the y-axis

On comparing the given equation with [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1

We get, a = 7 and b = 6

Therefore, c = √(a^{2} – b^{2}) = √(49-36) =√13

Thus, the coordinates of the foci are (√13, 0) and (-√13, 0)

The coordinates of the vertices are (7, 0) and (-7, 0)

Length of major axis = 2a = 14

Length of minor axis = 2b = 12

Eccentricity e = ^{c}/_{a} = ^{√13}/_{7}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 36}/_{7} = ^{72}/_{7}

**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse [(x**^{2})/_{100}] + [(y^{2})/_{400}]= 1

Solution:

The given equation is [(x^{2})/_{100}] + [(y^{2})/_{400}]= 1

Here the denominator of [(y^{2})/_{400}] is greater than the denominator of [(x^{2})/_{100}]

Therefore, the major axis is along the y- axis, while the minor axis is along the x-axis

On comparing the given equation with [(x^{2})/(b^{2})] + [(y^{2})/ (a^{2})]= 1

We get, a = 10 and b = 20

Therefore, c = √(a^{2} – b^{2}) = √(100 – 400) =√300 = ±10√3

Thus, the coordinates of the foci are (±10√3, 0) and (-10√3, 0)

The coordinates of the vertices are (20, 0) and (-20, 0)

Length of major axis = 2a = 40

Length of minor axis = 2b = 20

Eccentricity e = ^{c}/_{a} =^{10√3}/_{20} = ^{√3}/_{2}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 100}/_{20} = 10

**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse 36x**^{2}+ 4y^{2}= 1

Solution:

The given equation is 36x^{2} + 4y^{2}= 1

36x^{2} + 4y^{2}= 1 can also be written as [(x^{2})/_{4}] + [(y^{2})/_{36}] = 1

i.e., [(x^{2})/(2^{2})] + [(y^{2})/(6^{2})] = 1

Here the denominator of [(y^{2})/6^{2}] is greater than the denominator of [(x^{2})/2^{2}]

Therefore, the major axis is along the y- axis, while the minor axis is along the x-axis

On comparing the given equation with [(x^{2})/(b^{2})] + [(y^{2})/ (a^{2})]= 1

We get, a = 2 and b = 6

Therefore, c = √(a^{2} – b^{2}) = √(36-4) =√32 = 4√2

Thus, the coordinates of the foci are (4√2, 0) and (-4√2, 0)

The coordinates of the vertices are (6, 0) and (-6, 0)

Length of major axis = 2a = 12

Length of minor axis = 2b = 4

Eccentricity e = ^{c}/_{a} = ^{4√2}/_{6} = ^{2√2}/_{3}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 4}/_{6} = ^{4}/_{3}

**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse 16x**^{2}+ y^{2}= 16

Solution:

The given equation is 16x^{2} + y^{2}= 16

16x^{2} + y^{2}= 16 can also be written as [(x^{2})/_{1}] + [(y^{2})/_{16}] = 1

i.e., [(x^{2})/(1^{2})] + [(y^{2})/(4^{2})] = 1

Here the denominator of [(y^{2})/4^{2}] is greater than the denominator of [(x^{2})/1^{2}]

Therefore, the major axis is along the y- axis, while the minor axis is along the x-axis

On comparing the given equation with [(x^{2})/(b^{2})] + [(y^{2})/ (c^{2})]= 1

We get, a = 4 and b = 1

Therefore, c = √(a^{2} – b^{2}) = √(16-1) =√15

Thus, the coordinates of the foci are (√15, 0) and (-√15, 0)

The coordinates of the vertices are (4, 0) and (-4, 0)

Length of major axis = 2a = 8

Length of minor axis = 2b = 2

Eccentricity e = ^{c}/_{a} = ^{√15}/_{4}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 1}/_{4} = ^{1}/_{2}

**Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the rectum of the ellipse 4x**^{2}+ 9y^{2}= 36

Solution:

The given equation is 4x^{2} + 9y^{2}= 36

4x^{2} + 9y^{2}= 36 can also be written as [(x^{2})/_{9}] + [(y^{2})/_{4}] = 1

i.e., [(x^{2})/(3^{2})] + [(y^{2})/(2^{2})] = 1

Here the denominator of [(x^{2})/2^{2}] is greater than the denominator of [(y^{2})/3^{2}]

Therefore, the major axis is along the x- axis, while the minor axis is along the y-axis

On comparing the given equation with [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1

We get, a = 3 and b = 2

Therefore, c = √(a^{2} – b^{2}) = √(9-4) =√5

Thus, the coordinates of the foci are (√5, 0) and (-√5, 0)

The coordinates of the vertices are (3, 0) and (-3, 0)

Length of major axis = 2a = 6

Length of minor axis = 2b = 4

Eccentricity e = ^{c}/_{a} = ^{√5}/_{3} = ^{√5}/_{3}

Length of latus rectum = (2b^{2})/_{a} = ^{2 x 4}/_{3} = ^{8}/_{3}

**Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0)**

Solution:

Vertices (±5, 0), foci (±4, 0)

Here, the vertices are on the x-axis.

Therefore, the equation of the ellipse will be of the form [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1, where a is the semi-major axis.

Accordingly, a = 5 and c = 4.

It is known that a^{2} = b^{2} + c^{2}

5^{2} = b^{2} + 4^{2}

25 = b^{2} + 16

b^{2} = 25 – 16

b = √9 = 3

Thus, the equation of the ellipse is [(x^{2} )/(5^{2})]+ [(y^{2} )/(3^{2})] = 1 or [(x^{2} )/(25)]+ [(y^{2} )/(9)] = 1

**Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)**

Solution:

given vertices (0, ±13), foci (0, ±5)

Here, the vertices are on the y-axis.

Therefore, the equation of the ellipse will be of the form [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1, where a is the semi-major axis.

Accordingly, a = 13 and c = 5.

It is known that a^{2} = b^{2} + c^{2}

13^{2} = b^{2} + 5^{2}

169 = b^{2} + 25

b^{2} = 169 – 25

b = √144 = 12

Thus, the equation of the ellipse is [(x^{2} )/(12^{2})]+ [(y^{2} )/(13^{2})] = 1 or [(x^{2} )/(144)]+ [(y^{2} )/(169)] = 1

**Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0)**

Solution:

given vertices (±6, 0), foci (±4, 0)

Here, the vertices are on the x-axis.

Therefore, the equation of the ellipse will be of the form [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1, where a is the semi-major axis.

Accordingly, a = 6 and c = 4.

It is known that a^{2} = b^{2} + c^{2}

6^{2} = b^{2} + 4^{2}

36 = b^{2} + 16

b^{2} = 36 – 16

b = √20

Thus, the equation of the ellipse is [(x^{2} )/(6^{2})]+ [(y^{2} )/( √20)^{2}] = 1 or [(x^{2} )/36]+ [(y^{2} )/20] = 1

**Find the equation for the ellipse that satisfies the given conditions: Ends of major axis(±3, 0), ends of minor axis (0, ±2)**

solution;

given; ends of major axis (±3, 0), ends of minor axis (0, ±2)

Here, the major axis is along the x-axis.

Therefore, the equation of the ellipse will be of the form [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1 , where a is the semi-major axis.

Accordingly, a = 3 and b = 2.

Thus, the equation of the ellipse is

[(x^{2})/(3^{2})] + [(y^{2})/ (2^{2})]= 1 i.e., (x^{2})/_{9} + (y^{2})/_{4 }= 1

**Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (0, ±√5), ends of minor axis (±1, 0)**

solution:

Given ennds of major axis (0, ±√5), ends of minor axis (±1, 0)

Here, the major axis is along the y-axis.

^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1, where a is the semi-major axis.

Accordingly, a =√5 and b = 1.

Thus, the equation of the ellipse is

[(x^{2})/(1^{2})] + [(y^{2})/ (√5)^{2}]= 1

i.e.,

[(x^{2})/1] + [(y^{2})/ 5]= 1

**Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5, 0)**

solution:

Length of major axis = 26; foci = (±5, 0).

Since the foci are on the x-axis, the major axis is along the x-axis.

^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1, where a is the semi-major axis.

Accordingly, 2a = 26 ⇒ a = 13 and c = 5.

a^{2} = b^{2} + c^{2}

13^{2} = b^{2} + 5^{2}

b^{2} = 169 – 25

b = √144 = 12

Thus, the equation of the ellipse is [(x^{2})/(13^{2})] + [(y^{2})/ (12^{2})]= 1 i.e., [(x^{2})/_{169}] + [(y^{2})/ _{144}]= 1

**Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6)**

solution:

Length of minor axis = 16; foci = (0, ±6).

Since the foci are on the y-axis, the major axis is along the y-axis.

Therefore, the equation of the ellipse will be of the form [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1 , where a is the semi-major axis.

Accordingly, 2b = 16 ⇒ b = 8 and c = 6.

it is known that a^{2} = b^{2} + c^{2}

a^{2} = 8^{2}+6^{2}

a^{2 }= 64 + 36

a^{2 }= 100

a = 10

Thus, the equation of the ellipse is . [(x^{2})/(8^{2})] + [(y^{2})/ (10)^{2}]= 1 or [(x^{2})/_{64}] + [(y^{2})/_{100}]= 1

**Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), a = 4**

Solution:

Foci (±3, 0), a = 4

Since the foci are on the x-axis, the major axis is along the x-axis.

Therefore, the equation of the ellipse will be of the form [(x^{2})/(a^{2})] + [(y^{2})/ (b^{2})]= 1, where a is the

semi-major axis.

Accordingly, c = 3 and a = 4.

We know, a^{2} = b^{2} + c^{2}

4^{2} = b^{2} + 3^{2}

16 = b^{2} + 9

b^{2} = 16 – 9 = 7

Thus, the equation of the ellipse is [(x^{2})/(4^{2})] + [(y^{2})/ (√7)^{2}]= 1 or [(x^{2})/_{16}] + [(y^{2})/_{7}]= 1

**Find the equation for the ellipse that satisfies the given conditions: b = 3, c = 4, centre at the origin; foci on the x axis.**

Solution:

It is given that b = 3, c = 4, centre at the origin; foci on the x axis.

Since the foci are on the x-axis, the major axis is along the x-axis.

Therefore, the equation of the ellipse will be of the form [(x^{2})/(b^{2})] + [(y^{2})/(a)^{2}]= 1, where a is the semi-major axis.

Accordingly, b = 3, c = 4.

We know that, a^{2} = b^{2} + c^{2}

a^{2} = 3^{2} + 4^{2} = 9 = 16 = 25

a = 5

Thus, the equation of the ellipse is [(x^{2})/(5^{2})] + [(y^{2})/ (3)^{2}]= 1 or [(x^{2})/_{25}] + [(y^{2})/_{9}]= 1

**Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).**

solution:

Since the centre is at (0, 0) and the major axis is on the y-axis, the equation of the ellipse will be of the form

[(x^{2})/(b^{2})] + [(y^{2})/(a)^{2}]= 1 ———–(1)

where a is the semi-major axis

The ellipse passes through points (3, 2) and (1, 6).

Hence,

[(9^{2})/(b^{2})] + [(4^{2})/ (a)^{2}]= 1 ——–(2)

[^{1}/b^{2}] + [^{36}/a^{2}]= 1 —————-(3)

On solving equations (2) and (3), we obtain b^{2} = 10 and a^{2} = 40.

Thus, the equation of the ellipse is [(x^{2})/10] + [(y^{2})/40]= 1 or 4x^{2} + y^{2} = 40

**Find the equation for the ellipse that satisfies the given conditions: Major axis on the x -axis and passes through the points (4, 3) and (6, 2).**

Solution;

Since the major axis is on the x-axis, the equation of the ellipse will be of the form

(x^{2})/(a^{2}) + (y^{2})/(b^{2}) = 1 ———–(1), where a is the semi-major axis.

The ellipse passes through points (4, 3) and (6, 2). Hence,

[^{16}/(a^{2})] + [^{9}/(b)^{2}]= 1 ———–(2)

^{36}/(a^{2}) + ^{4}/(b^{2}) = 1 —————(3)

On solving equations (2) and (3), we obtain a^{2} = 52 and b^{2} = 13.

Thus, the equation of the ellipse is (x^{2})/_{52} + (y^{2})/_{13} = 1 or x^{2} + 4y^{2} = 52