- 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 [(x2)/36] + [(y2)/16]= 1
Solution:
The given equation is [(x2)/36] + [(y2)/16]= 1
Here the denominator of [(x2)/36] is greater than the denominator of [(y2)/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 [(x2)/(a2)] + [(y2)/ (b2)]= 1
Therefore, c = √(a2 – b2) = √(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 = (2b2)/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 [(x2)/4] + [(y2)/25]= 1
Solution:
The given equation is [(x2)/4] + [(y2)/25]= 1
Here the denominator of [(y2)/25] is greater than the denominator of [(x2)/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 [(x2)/(b2)] + [(y2)/ (a2)]= 1
We get, a = 5 and b = 2
c = √(a2 – b2) = √(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 = (2b2)/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 [(x2)/16] + [(y2)/9]= 1
Solution:
The given equation is [(x2)/16] + [(y2)/9]= 1
Here the denominator of [(x2)/16] is greater than the denominator of [(y2)/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 [(x2)/(a2)] + [(y2)/ (b2)]= 1
We get, a = 4 and b = 3
Therefore, c = √(a2 – b2) = √(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 = (2b2)/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 [(x2)/25] + [(y2)/100]= 1
Solution:
The given equation is [(x2)/25] + [(y2)/100]= 1
Here the denominator of [(y2)/100] is greater than the denominator of [(x2)/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 [(x2)/(b2)] + [(y2)/ (a2)]= 1
Therefore, c = √(a2 – b2) = √(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 = (2b2)/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 [(x2)/49] + [(y2)/36]= 1
Solution:
The given equation is [(x2)/49] + [(y2)/36]= 1
Here the denominator of [(x2)/49] is greater than the denominator of [(y2)/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 [(x2)/(a2)] + [(y2)/ (b2)]= 1
We get, a = 7 and b = 6
Therefore, c = √(a2 – b2) = √(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 = (2b2)/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 [(x2)/100] + [(y2)/400]= 1
Solution:
The given equation is [(x2)/100] + [(y2)/400]= 1
Here the denominator of [(y2)/400] is greater than the denominator of [(x2)/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 [(x2)/(b2)] + [(y2)/ (a2)]= 1
We get, a = 10 and b = 20
Therefore, c = √(a2 – b2) = √(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 = (2b2)/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 36x2 + 4y2= 1
Solution:
The given equation is 36x2 + 4y2= 1
36x2 + 4y2= 1 can also be written as [(x2)/4] + [(y2)/36] = 1
i.e., [(x2)/(22)] + [(y2)/(62)] = 1
Here the denominator of [(y2)/62] is greater than the denominator of [(x2)/22]
Therefore, the major axis is along the y- axis, while the minor axis is along the x-axis
On comparing the given equation with [(x2)/(b2)] + [(y2)/ (a2)]= 1
We get, a = 2 and b = 6
Therefore, c = √(a2 – b2) = √(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 = (2b2)/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 16x2 + y2= 16
Solution:
The given equation is 16x2 + y2= 16
16x2 + y2= 16 can also be written as [(x2)/1] + [(y2)/16] = 1
i.e., [(x2)/(12)] + [(y2)/(42)] = 1
Here the denominator of [(y2)/42] is greater than the denominator of [(x2)/12]
Therefore, the major axis is along the y- axis, while the minor axis is along the x-axis
On comparing the given equation with [(x2)/(b2)] + [(y2)/ (c2)]= 1
We get, a = 4 and b = 1
Therefore, c = √(a2 – b2) = √(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 = (2b2)/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 4x2 + 9y2= 36
Solution:
The given equation is 4x2 + 9y2= 36
4x2 + 9y2= 36 can also be written as [(x2)/9] + [(y2)/4] = 1
i.e., [(x2)/(32)] + [(y2)/(22)] = 1
Here the denominator of [(x2)/22] is greater than the denominator of [(y2)/32]
Therefore, the major axis is along the x- axis, while the minor axis is along the y-axis
On comparing the given equation with [(x2)/(a2)] + [(y2)/ (b2)]= 1
We get, a = 3 and b = 2
Therefore, c = √(a2 – b2) = √(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 = (2b2)/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 [(x2)/(a2)] + [(y2)/ (b2)]= 1, where a is the semi-major axis.
Accordingly, a = 5 and c = 4.
It is known that a2 = b2 + c2
52 = b2 + 42
25 = b2 + 16
b2 = 25 – 16
b = √9 = 3
Thus, the equation of the ellipse is [(x2 )/(52)]+ [(y2 )/(32)] = 1 or [(x2 )/(25)]+ [(y2 )/(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 [(x2)/(a2)] + [(y2)/ (b2)]= 1, where a is the semi-major axis.
Accordingly, a = 13 and c = 5.
It is known that a2 = b2 + c2
132 = b2 + 52
169 = b2 + 25
b2 = 169 – 25
b = √144 = 12
Thus, the equation of the ellipse is [(x2 )/(122)]+ [(y2 )/(132)] = 1 or [(x2 )/(144)]+ [(y2 )/(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 [(x2)/(a2)] + [(y2)/ (b2)]= 1, where a is the semi-major axis.
Accordingly, a = 6 and c = 4.
It is known that a2 = b2 + c2
62 = b2 + 42
36 = b2 + 16
b2 = 36 – 16
b = √20
Thus, the equation of the ellipse is [(x2 )/(62)]+ [(y2 )/( √20)2] = 1 or [(x2 )/36]+ [(y2 )/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 [(x2)/(a2)] + [(y2)/ (b2)]= 1 , where a is the semi-major axis.
Accordingly, a = 3 and b = 2.
Thus, the equation of the ellipse is
[(x2)/(32)] + [(y2)/ (22)]= 1 i.e., (x2)/9 + (y2)/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.
Therefore, the equation of the ellipse will be of the form [(x2)/(a2)] + [(y2)/ (b2)]= 1, where a is the semi-major axis.
Accordingly, a =√5 and b = 1.
Thus, the equation of the ellipse is
[(x2)/(12)] + [(y2)/ (√5)2]= 1
i.e.,
[(x2)/1] + [(y2)/ 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.
Therefore, the equation of the ellipse will be of the form [(x2)/(a2)] + [(y2)/ (b2)]= 1, where a is the semi-major axis.
Accordingly, 2a = 26 ⇒ a = 13 and c = 5.
a2 = b2 + c2
132 = b2 + 52
b2 = 169 – 25
b = √144 = 12
Thus, the equation of the ellipse is [(x2)/(132)] + [(y2)/ (122)]= 1 i.e., [(x2)/169] + [(y2)/ 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 [(x2)/(a2)] + [(y2)/ (b2)]= 1 , where a is the semi-major axis.
Accordingly, 2b = 16 ⇒ b = 8 and c = 6.
it is known that a2 = b2 + c2
a2 = 82+62
a2 = 64 + 36
a2 = 100
a = 10
Thus, the equation of the ellipse is . [(x2)/(82)] + [(y2)/ (10)2]= 1 or [(x2)/64] + [(y2)/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 [(x2)/(a2)] + [(y2)/ (b2)]= 1, where a is the
semi-major axis.
Accordingly, c = 3 and a = 4.
We know, a2 = b2 + c2
42 = b2 + 32
16 = b2 + 9
b2 = 16 – 9 = 7
Thus, the equation of the ellipse is [(x2)/(42)] + [(y2)/ (√7)2]= 1 or [(x2)/16] + [(y2)/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 [(x2)/(b2)] + [(y2)/(a)2]= 1, where a is the semi-major axis.
Accordingly, b = 3, c = 4.
We know that, a2 = b2 + c2
a2 = 32 + 42 = 9 = 16 = 25
a = 5
Thus, the equation of the ellipse is [(x2)/(52)] + [(y2)/ (3)2]= 1 or [(x2)/25] + [(y2)/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
[(x2)/(b2)] + [(y2)/(a)2]= 1 ———–(1)
where a is the semi-major axis
The ellipse passes through points (3, 2) and (1, 6).
Hence,
[(92)/(b2)] + [(42)/ (a)2]= 1 ——–(2)
[1/b2] + [36/a2]= 1 —————-(3)
On solving equations (2) and (3), we obtain b2 = 10 and a2 = 40.
Thus, the equation of the ellipse is [(x2)/10] + [(y2)/40]= 1 or 4x2 + y2 = 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
(x2)/(a2) + (y2)/(b2) = 1 ———–(1), where a is the semi-major axis.
The ellipse passes through points (4, 3) and (6, 2). Hence,
[16/(a2)] + [9/(b)2]= 1 ———–(2)
36/(a2) + 4/(b2) = 1 —————(3)
On solving equations (2) and (3), we obtain a2 = 52 and b2 = 13.
Thus, the equation of the ellipse is (x2)/52 + (y2)/13 = 1 or x2 + 4y2 = 52
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