Question -
Answer -
(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.