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 Page 1


26. Ellipse
Exercise 26.1
1. Question
Find the equation of the ellipse whose focus is (1, - 2), the directrix 3x – 2y + 5 = 0 and eccentricity equal to
1/2.
Answer
Given that we need to find the equation of the ellipse whose focus is S(1, - 2) and directrix(M) is 3x - 2y + 5
= 0 and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 52x
2
 + 52y
2
 - 104x + 208y + 260 = 9x
2
 + 4y
2
 - 12xy - 20y + 30x + 25
? 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0
? The equation of the ellipse is 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0.
2 A. Question
Find the equation of the ellipse in the following cases:
focus is (0, 1), directrix is x + y = 0 and .
Answer
Page 2


26. Ellipse
Exercise 26.1
1. Question
Find the equation of the ellipse whose focus is (1, - 2), the directrix 3x – 2y + 5 = 0 and eccentricity equal to
1/2.
Answer
Given that we need to find the equation of the ellipse whose focus is S(1, - 2) and directrix(M) is 3x - 2y + 5
= 0 and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 52x
2
 + 52y
2
 - 104x + 208y + 260 = 9x
2
 + 4y
2
 - 12xy - 20y + 30x + 25
? 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0
? The equation of the ellipse is 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0.
2 A. Question
Find the equation of the ellipse in the following cases:
focus is (0, 1), directrix is x + y = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(0,1) and directrix(M) is x + y = 0 and
eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 8x
2
 + 8y
2
 - 16y + 8 = x
2
 + y
2
 + 2xy
? 7x
2
 + 7y
2
 - 2xy - 16y + 8 = 0
? The equation of the ellipse is 7x
2
 + 7y
2
 - 2xy - 16y + 8 = 0.
2 B. Question
Find the equation of the ellipse in the following cases:
focus is (- 1, 1), directrix is x - y + 3 = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(- 1,1) and directrix(M) is x - y + 3 = 0
and eccentricity(e) is equal to .
Page 3


26. Ellipse
Exercise 26.1
1. Question
Find the equation of the ellipse whose focus is (1, - 2), the directrix 3x – 2y + 5 = 0 and eccentricity equal to
1/2.
Answer
Given that we need to find the equation of the ellipse whose focus is S(1, - 2) and directrix(M) is 3x - 2y + 5
= 0 and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 52x
2
 + 52y
2
 - 104x + 208y + 260 = 9x
2
 + 4y
2
 - 12xy - 20y + 30x + 25
? 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0
? The equation of the ellipse is 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0.
2 A. Question
Find the equation of the ellipse in the following cases:
focus is (0, 1), directrix is x + y = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(0,1) and directrix(M) is x + y = 0 and
eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 8x
2
 + 8y
2
 - 16y + 8 = x
2
 + y
2
 + 2xy
? 7x
2
 + 7y
2
 - 2xy - 16y + 8 = 0
? The equation of the ellipse is 7x
2
 + 7y
2
 - 2xy - 16y + 8 = 0.
2 B. Question
Find the equation of the ellipse in the following cases:
focus is (- 1, 1), directrix is x - y + 3 = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(- 1,1) and directrix(M) is x - y + 3 = 0
and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 8x
2
 + 8y
2
 + 16x - 16y + 16 = x
2
 + y
2
 - 2xy + 6x - 6y + 9
? 7x
2
 + 7y
2
 + 2xy + 10x - 10y + 7 = 0
? The equation of the ellipse is 7x
2
 + 7y
2
 + 2xy + 10x - 10y + 7 = 0.
2 C. Question
Find the equation of the ellipse in the following cases:
focus is (- 2, 3), directrix is 2x + 3y + 4 = 0 and 
Answer
Given that we need to find the equation of the ellipse whose focus is S(- 2,3) and directrix(M) is 2x + 3y + 4
= 0 and eccentricity(e) is equal to .
Page 4


26. Ellipse
Exercise 26.1
1. Question
Find the equation of the ellipse whose focus is (1, - 2), the directrix 3x – 2y + 5 = 0 and eccentricity equal to
1/2.
Answer
Given that we need to find the equation of the ellipse whose focus is S(1, - 2) and directrix(M) is 3x - 2y + 5
= 0 and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 52x
2
 + 52y
2
 - 104x + 208y + 260 = 9x
2
 + 4y
2
 - 12xy - 20y + 30x + 25
? 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0
? The equation of the ellipse is 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0.
2 A. Question
Find the equation of the ellipse in the following cases:
focus is (0, 1), directrix is x + y = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(0,1) and directrix(M) is x + y = 0 and
eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 8x
2
 + 8y
2
 - 16y + 8 = x
2
 + y
2
 + 2xy
? 7x
2
 + 7y
2
 - 2xy - 16y + 8 = 0
? The equation of the ellipse is 7x
2
 + 7y
2
 - 2xy - 16y + 8 = 0.
2 B. Question
Find the equation of the ellipse in the following cases:
focus is (- 1, 1), directrix is x - y + 3 = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(- 1,1) and directrix(M) is x - y + 3 = 0
and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 8x
2
 + 8y
2
 + 16x - 16y + 16 = x
2
 + y
2
 - 2xy + 6x - 6y + 9
? 7x
2
 + 7y
2
 + 2xy + 10x - 10y + 7 = 0
? The equation of the ellipse is 7x
2
 + 7y
2
 + 2xy + 10x - 10y + 7 = 0.
2 C. Question
Find the equation of the ellipse in the following cases:
focus is (- 2, 3), directrix is 2x + 3y + 4 = 0 and 
Answer
Given that we need to find the equation of the ellipse whose focus is S(- 2,3) and directrix(M) is 2x + 3y + 4
= 0 and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 325x
2
 + 325y
2
 + 1300x - 1950y + 4225 = 64x
2
 + 144y
2
 + 192xy + 256x + 384y + 256
? 261x
2
 + 181y
2
 - 192xy + 1044x - 2334y + 3969 = 0
? The equation of the ellipse is 261x
2
 + 181y
2
 - 192xy + 1044x - 2334y + 3969 = 0.
2 D. Question
Find the equation of the ellipse in the following cases:
focus is (1, 2), directrix is 3x + 4y - 7 = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(1, 2) and directrix(M) is 3x + 4y - 5 =
0 and eccentricity(e) is equal to .
Page 5


26. Ellipse
Exercise 26.1
1. Question
Find the equation of the ellipse whose focus is (1, - 2), the directrix 3x – 2y + 5 = 0 and eccentricity equal to
1/2.
Answer
Given that we need to find the equation of the ellipse whose focus is S(1, - 2) and directrix(M) is 3x - 2y + 5
= 0 and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 52x
2
 + 52y
2
 - 104x + 208y + 260 = 9x
2
 + 4y
2
 - 12xy - 20y + 30x + 25
? 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0
? The equation of the ellipse is 43x
2
 + 48y
2
 + 12xy - 134x + 228y + 235 = 0.
2 A. Question
Find the equation of the ellipse in the following cases:
focus is (0, 1), directrix is x + y = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(0,1) and directrix(M) is x + y = 0 and
eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 8x
2
 + 8y
2
 - 16y + 8 = x
2
 + y
2
 + 2xy
? 7x
2
 + 7y
2
 - 2xy - 16y + 8 = 0
? The equation of the ellipse is 7x
2
 + 7y
2
 - 2xy - 16y + 8 = 0.
2 B. Question
Find the equation of the ellipse in the following cases:
focus is (- 1, 1), directrix is x - y + 3 = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(- 1,1) and directrix(M) is x - y + 3 = 0
and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 8x
2
 + 8y
2
 + 16x - 16y + 16 = x
2
 + y
2
 - 2xy + 6x - 6y + 9
? 7x
2
 + 7y
2
 + 2xy + 10x - 10y + 7 = 0
? The equation of the ellipse is 7x
2
 + 7y
2
 + 2xy + 10x - 10y + 7 = 0.
2 C. Question
Find the equation of the ellipse in the following cases:
focus is (- 2, 3), directrix is 2x + 3y + 4 = 0 and 
Answer
Given that we need to find the equation of the ellipse whose focus is S(- 2,3) and directrix(M) is 2x + 3y + 4
= 0 and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 325x
2
 + 325y
2
 + 1300x - 1950y + 4225 = 64x
2
 + 144y
2
 + 192xy + 256x + 384y + 256
? 261x
2
 + 181y
2
 - 192xy + 1044x - 2334y + 3969 = 0
? The equation of the ellipse is 261x
2
 + 181y
2
 - 192xy + 1044x - 2334y + 3969 = 0.
2 D. Question
Find the equation of the ellipse in the following cases:
focus is (1, 2), directrix is 3x + 4y - 7 = 0 and .
Answer
Given that we need to find the equation of the ellipse whose focus is S(1, 2) and directrix(M) is 3x + 4y - 5 =
0 and eccentricity(e) is equal to .
Let P(x,y) be any point on the ellipse.
We know that the distance between the focus and any point on the ellipse is equal to the eccentricity times
the perpendicular distance from that point to the directrix.
We know that distance between the points (x
1
,y
1
) and (x
2
,y
2
) is .
We know that the perpendicular distance from the point (x
1
,y
1
) to the line ax + by + c = 0 is .
? SP = ePM
? SP
2
 = e
2
PM
2
? 
? 
? 
? 100x
2
 + 100y
2
 - 200x - 400y + 500 = 9x
2
 + 16y
2
 + 24xy - 30x - 40y + 25
? 91x
2
 + 84y
2
 - 24xy - 170x - 360y + 475 = 0
? The equation of the ellipse is 91x
2
 + 84y
2
 - 24xy - 170x - 360y + 475 = 0.
3 A. Question
Find the eccentricity, coordinates of foci, length of the latus - rectum of the following ellipse:
4x
2
 + 9y
2
 = 1
Answer
Given the equation of the ellipse is 4x
2
 + 9y
2
 = 1.
We need to find the eccentricity, coordinates of foci and length of latus rectum.

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