RD Chapter 26 Ellipse Solutions
Question - 1 : - Find the equation of the ellipse whose focus is (1, -2), the directrix 3x – 2y + 5 = 0 and eccentricity equal to 1/2.
Answer - 1 : -
Given:
Focus = (1, -2)
Directrix = 3x – 2y + 5 = 0
Eccentricity = ½
Let P(x, y) be any point on the ellipse.
We know that distance between the points (x1, y1)and (x2, y2) is given as
We also know that the perpendicular distance from the point(x1, y1) to the line ax + by + c = 0 is given as
So,
SP = ePM
SP2 = e2PM2
Upon cross multiplying, we get
52x2 + 52y2 – 104x + 208y +260 = 9x2 + 4y2 – 12xy – 20y + 30x + 25
43x2 + 48y2 + 12xy – 134x +228y + 235 = 0
∴ The equation of the ellipse is 43x2 +48y2 + 12xy – 134x + 228y + 235 = 0
Question - 2 : - Find the equation of the ellipse in the following cases:
(i) focus is (0, 1), directrix is x + y = 0 and e = ½.
(ii) focus is (- 1, 1), directrix is x – y + 3 = 0 and e = ½.
(iii) focus is (- 2, 3), directrix is 2x + 3y + 4 = 0 and e = 4/5.
(iv) focus is (1, 2), directrix is 3x + 4y – 7 = 0 and e = ½.
Answer - 2 : -
(i) focus is (0, 1), directrix is x + y = 0 and e = ½
Given:
Focus is (0, 1)
Directrix is x + y = 0
e = ½
Let P(x, y) be any point on the ellipse.
We know that distance between the points (x1, y1)and (x2, y2) is given as
We also know that the perpendicular distance from the point(x1, y1) to the line ax + by + c = 0 is given as
So,
SP = ePM
SP2 = e2PM2
Upon cross multiplying, we get
8x2 + 8y2 – 16y + 8 = x2 +y2 + 2xy
7x2 + 7y2 – 2xy – 16y + 8 =0
∴ The equation of the ellipse is 7x2 +7y2 – 2xy – 16y + 8 = 0
(ii) focus is (- 1, 1), directrix is x – y + 3 = 0 and e = ½
Given:
Focus is (- 1, 1)
Directrix is x – y + 3 = 0
e = ½
Let P(x, y) be any point on the ellipse.
We know that distance between the points (x1, y1)and (x2, y2) is given as
We also know that the perpendicular distance from the point(x1, y1) to the line ax + by + c = 0 is given as
So,
SP = ePM
SP2 = e2PM2
Upon cross multiplying, we get
8x2 + 8y2 + 16x – 16y + 16 =x2 + y2 – 2xy + 6x – 6y + 9
7x2 + 7y2 + 2xy + 10x – 10y+ 7 = 0
∴ The equation of the ellipse is 7x2 +7y2 + 2xy + 10x – 10y + 7 = 0
(iii) focus is (- 2, 3), directrix is 2x + 3y + 4 = 0 and e = 4/5
Focus is (- 2, 3)
Directrix is 2x + 3y + 4 = 0
e = 4/5
Let P(x, y) be any point on the ellipse.
We know that distance between the points (x1, y1)and (x2, y2) is given as
We also know that the perpendicular distance from the point(x1, y1) to the line ax + by + c = 0 is given as
So,
SP = ePM
SP2 = e2PM2
Upon cross multiplying, we get
325x2 + 325y2 + 1300x –1950y + 4225 = 64x2 + 144y2 + 192xy + 256x +384y + 256
261x2 + 181y2 – 192xy +1044x – 2334y + 3969 = 0
∴ The equation of the ellipse is 261x2 +181y2 – 192xy + 1044x – 2334y + 3969 = 0
(iv) focus is (1, 2), directrix is 3x + 4y – 7 = 0 and e =½.
Given:
focus is (1, 2)
directrix is 3x + 4y – 7 = 0
e = ½.
Let P(x, y) be any point on the ellipse.
We know that distance between the points (x1, y1)and (x2, y2) is given as
We also know that the perpendicular distance from the point(x1, y1) to the line ax + by + c = 0 is given as
So,
SP = ePM
SP2 = e2PM2
Upon cross multiplying, we get
100x2 + 100y2 – 200x – 400y+ 500 = 9x2 + 16y2 + 24xy – 30x – 40y + 25
91x2 + 84y2 – 24xy – 170x –360y + 475 = 0
∴ The equation of the ellipse is 91x2 +84y2 – 24xy – 170x – 360y + 475 = 0
Question - 3 : - Find the eccentricity, coordinatesof foci, length of the latus – rectum of the following ellipse:
(i) 4x2 + 9y2 = 1
(ii) 5x2 + 4y2 =1
(iii) 4x2 + 3y2 =1
(iv) 25x2 + 16y2 =1600
(v) 9x2 + 25y2 =225
Answer - 3 : -
(i) 4x2 + 9y2 = 1
Given:
The equation of ellipse => 4x2 + 9y2 =1
This equation can be expressed as
By using the formula,
Eccentricity:
Here, a2 = ¼, b2 = 1/9
Length of latus rectum = 2b2/a
= [2 (1/9)] / (1/2)
= 4/9
Coordinates of foci (±ae, 0)
(ii) 5x2 + 4y2 = 1
Given:
The equation of ellipse => 5x2 + 4y2 =1
This equation can be expressed as
By using the formula,
Eccentricity:
Here, a2 = 1/5 and b2 = ¼
Length of latus rectum = 2b2/a
= [2(1/5)] / (1/2)
= 4/5
Coordinates of foci (±ae, 0)
(iii) 4x2 + 3y2 = 1
Given:
The equation of ellipse => 4x2 + 3y2 =1
This equation can be expressed as
By using the formula,
Eccentricity:
Here, a2 = 1/4 and b2 = 1/3
Length of latus rectum = 2b2/a
= [2(1/4)] / (1/√3)
= √3/2
Coordinates of foci (±ae, 0)
(iv) 25x2 + 16y2 = 1600
Given:
The equation of ellipse => 25x2 + 16y2 =1600
This equation can be expressed as
By using the formula,
Eccentricity:
Here, a2 = 64 and b2 = 100
Length of latus rectum = 2b2/a
= [2(64)] / (100)
= 32/25
Coordinates of foci (±ae, 0)
(v) 9x2 + 25y2 = 225
Given:
The equation of ellipse => 9x2 + 25y2 =225
This equation can be expressed as
By using the formula,
Eccentricity:
Here, a2 = 25 and b2 = 9
Length of latus rectum = 2b2/a
= [2(9)] / (5)
= 18/5
Coordinates of foci (±ae, 0)
Question - 4 : - Find the equation to the ellipse (referred to its axes as the axes of x and y respectively) which passes through the point (-3, 1) and has eccentricity √(2/5).
Answer - 4 : -
Given:
The point (-3, 1)
Eccentricity = √(2/5)
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are xand y – axis is given as
…. (1)
Now let us substitute equation (2) in equation (1), we get
It is given that the curve passes through the point (-3, 1).
So by substituting the point in the curve we get,
3(- 3)2 + 5(1)2 = 3a2
3(9) + 5 = 3a2
32 = 3a2
a2 = 32/3
From equation (2)
b2 = 3a2/5
= 3(32/3) / 5
= 32/5
So now, the equation of the ellipse is given as:
3x2 + 5y2 = 32
∴ The equation of the ellipse is 3x2 +5y2 = 32.
Question - 5 : - Find the equation of the ellipse in the following cases:
(i) eccentricity e = ½ and foci (± 2, 0)
(ii) eccentricity e = 2/3 and length of latus – rectum = 5
(iii) eccentricity e = ½ and semi – major axis = 4
(iv) eccentricity e = ½ and major axis = 12
(v) The ellipse passes through (1, 4) and (- 6, 1)
Answer - 5 : -
(i) Eccentricity e = ½ and foci (± 2, 0)
Given:
Eccentricity e = ½
Foci (± 2, 0)
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are xand y – axis is given as
By using the formula,
Eccentricity:
b2 = 3a2/4
It is given that foci (± 2, 0) =>foci = (±ae, 0)
Where, ae = 2
a(1/2) = 2
a = 4
a2 = 16
We know b2 = 3a2/4
b2 = 3(16)/4
= 12
So the equation of the ellipse can be given as
3x2 + 4y2 = 48
∴ The equation of the ellipse is 3x2 +4y2 = 48
(ii) eccentricity e = 2/3 and length of latus rectum = 5
Given:
Eccentricity e = 2/3
Length of latus – rectum = 5
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are xand y – axis is given as
By using the formula,
Eccentricity:
By using the formula, length of the latus rectum is 2b2/a
So the equation of the ellipse can be given as
20x2 + 36y2 = 405
∴ The equation of the ellipse is 20x2 +36y2 = 405.
(iii) eccentricity e = ½ andsemi – major axis = 4
Given:
Eccentricity e = ½
Semi – major axis = 4
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are xand y – axis is given as
By using the formula,
Eccentricity:
It is given that the length of the semi – major axis is a
a = 4
a2 = 16
We know, b2 = 3a2/4
b2 = 3(16)/4
= 4
So the equation of the ellipse can be given as
3x2 + 4y2 = 48
∴ The equation of the ellipse is 3x2 +4y2 = 48.
(iv) eccentricity e = ½ and major axis = 12
Given:
Eccentricity e = ½
Major axis = 12
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are xand y – axis is given as
By using the formula,
Eccentricity:
b2 = 3a2/4
It is given that length of major axis is 2a.
2a = 12
a = 6
a2 = 36
So, by substituting the value of a2, we get
b2 = 3(36)/4
= 27
So the equation of the ellipse can be given as
3x2 + 4y2 = 108
∴ The equation of the ellipse is 3x2 +4y2 = 108.
(v) The ellipse passes through (1, 4) and (- 6, 1)
Given:
The points (1, 4) and (- 6, 1)
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are xand y – axis is given as
…. (1)
Let us substitute the point (1, 4) in equation (1), we get
b2 + 16a2 = a2 b2 ….(2)
Let us substitute the point (-6, 1) in equation (1), we get
a2 + 36b2 = a2b2 ….(3)
Let us multiply equation (3) by 16 and subtract withequation (2), we get
(16a2 + 576b2) – (b2 +16a2) = (16a2b2 – a2b2)
575b2 = 15a2b2
15a2 = 575
a2 = 575/15
= 115/3
So from equation (2),
So the equation of the ellipse can be given as
3x2 + 7y2 = 115
∴ The equation of the ellipse is 3x2 +7y2 = 115.
Question - 6 : - Find the equation of the ellipse whose foci are (4, 0) and (- 4, 0), eccentricity = 1/3.
Answer - 6 : -
Given:
Foci are (4, 0) (- 4, 0)
Eccentricity = 1/3.
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are xand y – axis is given as
By using the formula,
Eccentricity:
It is given that foci = (4, 0) (- 4, 0) => foci = (±ae,0)
Where, ae = 4
a(1/3) = 4
a = 12
a2 = 144
By substituting the value of a2, we get
b2 = 8a2/9
b2 = 8(144)/9
= 128
So the equation of the ellipse can be given as
Question - 7 : - Find the equation of the ellipse in the standard form whose minor axis is equal to the distance between foci and whose latus – rectum is 10.
Answer - 7 : -
Given:
Minor axis is equal to the distance between foci and whoselatus – rectum is 10.
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are xand y – axis is given as
We know that length of the minor axis is 2b and distancebetween the foci is 2ae.
By using the formula,
Eccentricity:
We know that the length of the latus rectum is 2b2/a
It is given that length of the latus rectum = 10
So by equating, we get
2b2/a = 10
a2/ a = 10 [Since, a2 = 2b2]
a = 10
a2 = 100
Now, by substituting the value of a2 we get
2b2/a = 10
2b2/10 = 10
2b2 = 10(10)
b2 = 100/2
= 50
So the equation of the ellipse can be given as
x2 + 2y2 = 100
∴ The equation of the ellipse is x2 + 2y2 =100.
Question - 8 : - Find the equation of the ellipse whose centre is (-2, 3) and whose semi – axis are 3 and 2 when the major axis is (i) parallel to x – axis (ii) parallel to the y – axis.
Answer - 8 : -
Given:
Centre = (-2, 3)
Semi – axis are 3 and 2
(i) Whenmajor axis is parallel to x-axis
Now let us find the equation to the ellipse.
We know that the equation of the ellipse with centre (p, q)is given by
Since major axis is parallel to x – axis
So, a = 3 and b = 2.
a2 = 9
b2 = 4
So the equation of the ellipse can be given as
4(x2 + 4x + 4) + 9(y2 – 6y +9) = 36
4x2 + 16x + 16 + 9y2 – 54y +81 = 36
4x2 + 9y2 + 16x – 54y + 61 =0
∴ The equation of the ellipse is 4x2 +9y2 + 16x – 54y + 61 = 0.
(ii) Whenmajor axis is parallel to y-axis
Now let us find the equation to the ellipse.
We know that the equation of the ellipse with centre (p, q)is given by
Since major axis is parallel to y – axis
So, a = 2 and b = 3.
a2 = 4
b2 = 9
So the equation of the ellipse can be given as
9(x2 + 4x + 4) + 4(y2 – 6y +9) = 36
9x2 + 36x + 36 + 4y2 – 24y +36 = 36
9x2 + 4y2 + 36x – 24y + 36 =0
∴ The equation of the ellipse is 9x2 +4y2 + 36x – 24y + 36 = 0.
Question - 9 : - Find the eccentricity of an ellipse whose latus – rectum is
(i) Half of its minor axis
(ii) Half of its major axis
Answer - 9 : -
Given:
We need to find the eccentricity of an ellipse.
(i) Iflatus – rectum is half of its minor axis
We know that the length of the semi – minor axis is b andthe length of the latus – rectum is 2b2/a.
2b2/a = b
a = 2b …. (1)
By using the formula,
We know that eccentricity of an ellipse is given as
(ii) Iflatus – rectum is half of its major axis
We know that the length of the semi – major axis is a andthe length of the latus – rectum is 2b2/a.
2b2/a
a2 = 2b2 …. (1)
By using the formula,
We know that eccentricity of an ellipse is given as
