NCERT Solutions: Exercise 10.3 - Conic Sections

Q1: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 
Ans: The given equation is .
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with , we obtain a = 6 and b = 4.
 
Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (6, 0) and (-6, 0).
Length of major axis = 2a = 12
Length of minor axis = 2b = 8

 
Length of latus rectum

Q2: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 
Ans: The given equation is .
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with , we obtain b = 2 and a = 5.

 
Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (0, 5) and (0, -5)
Length of major axis = 2a = 10
Length of minor axis = 2b = 4

 
Length of latus rectum

Q3: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 
Ans: The given equation is .
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with , we obtain a = 4 and b = 3.

Therefore,
The coordinates of the foci are .
The coordinates of the vertices are .
Length of major axis = 2a = 8
Length of minor axis = 2b = 6

 
Length of latus rectum

Q4: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 
Ans: The given equation is .
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with , we obtain b = 5 and a = 10.

Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (0, ±10).
Length of major axis = 2a = 20
Length of minor axis = 2b = 10

 
Length of latus rectum

Q5: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 
Ans: The given equation is .
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with , we obtain a = 7 and b = 6.

 
Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (± 7, 0).
Length of major axis = 2a = 14
Length of minor axis = 2b = 12

 
Length of latus rectum

Q6: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 
Ans: The given equation is .
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with , we obtain b = 10 and a = 20.

 
Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (0, ±20)
Length of major axis = 2a = 40
Length of minor axis = 2b = 20

 
Length of latus rectum

Q7: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 36x² + 4y² = 144
Ans: The given equation is 36x² + 4y² = 144.
It can be written as

 
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing equation (1) with , we obtain b = 2 and a = 6.

 
Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (0, ±6).
Length of major axis = 2a = 12
Length of minor axis = 2b = 4

 
Length of latus rectum

Q8: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 16x² +  y² = 16
Ans: The given equation is 16x² +  y² = 16.
It can be written as

 
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing equation (1) with , we obtain b = 1 and a = 4.

 
Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (0, ±4).
Length of major axis = 2a = 8
Length of minor axis = 2b = 2

 
Length of latus rectum

Q9: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 4x² + 9y² = 36
Ans: The given equation is 4x² + 9y² = 36.
It can be written as

 
Here, the denominator of  is greater than the denominator of .
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with , we obtain a = 3 and b = 2.

Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (±3, 0).
Length of major axis = 2a = 6
Length of minor axis = 2b = 4

 
Length of latus rectum

Q10: Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0)
Ans: 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 , where a is the semi-major axis.
Accordingly, a = 5 and c = 4.
It is known that .

 
Thus, the equation of the ellipse is .

Q11: Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)
Ans: 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 , where a is the semi-major axis.
Accordingly, a = 13 and c = 5.
It is known that .

Thus, the equation of the ellipse is .

Q12: Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0)
Ans: 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 , where a is the semi-major axis.
Accordingly, a = 6, c = 4.
It is known that .

 
Thus, the equation of the ellipse is .

Q13: Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (±3, 0), ends of minor axis (0, ±2)
Ans: 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 , where a is the semi-major axis.
Accordingly, a = 3 and b = 2.
Thus, the equation of the ellipse is .

Q14: Find the equation for the ellipse that satisfies the given conditions: Ends of major axis , ends of minor axis (±1, 0)
Ans: Ends of major axis , 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 , where a is the semi-major axis.
Accordingly, a =   and b = 1.
Thus, the equation of the ellipse is .

Q15: Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5, 0)
Ans: 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 , where a is the semi-major axis.
Accordingly, 2a = 26 ⇒ a = 13 and c = 5.
It is known that .

Thus, the equation of the ellipse is .

Q16: Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6)
Ans: 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 , where a is the semi-major axis.
Accordingly, 2b = 16 ⇒ b = 8 and c = 6.
It is known that .

 
Thus, the equation of the ellipse is .

Q17: Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), a = 4
Ans: 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 , where a is the semi-major axis.
Accordingly, c = 3 and a = 4.
It is known that .

Thus, the equation of the ellipse is .

Q18: Find the equation for the ellipse that satisfies the given conditions: b = 3, c = 4, centre at the origin; foci on the x axis.
Ans: 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 , where a is the semi-major axis.
Accordingly, b = 3, c = 4.
It is known that .

 
Thus, the equation of the ellipse is .

Q19: 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).
Ans: 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

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

 
On solving equations (2) and (3), we obtain b² = 10 and a² = 40.
Thus, the equation of the ellipse is .

Q20: 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).
Ans: Since the major axis is on the x-axis, the equation of the ellipse will be of the form

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

 
On solving equations (2) and (3), we obtain a² = 52 and b² = 13.
Thus, the equation of the ellipse is .