Equation of an ellipse is x2+ y2+ xy + 1 = 0.
(1) Length of largest chord = ?
(2) Length of chord perpendicular to the chord asked in (1) , passing through the center of ellipse = ?
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SOLUTION:--
First of all the correct equation of ellipse should be x2 + y2 + xy - 1 = 0
The constant 1 appears with positive sign in RHS
We will trace the shape and axes of ellipse
Let f(x,y) = x2 + y2 + xy - 1
(i). We notice that on interchanging x with y, the equation remains unchanged
f(x, y) = f(y, x)
x2 + y2 + xy - 1 = y2 + x2 + yx - 1
Thus the ellipse is symmetric about the line y = x
(ii). Also f(x,y) = f(-y, -x)
The ellipse is symmetric about the line y = -x
From this we conclude that it is an oblique ellipse whose centre is at origin and axes lying along y = x and y = -x
Let us find the coordinates of intersection of ellipse with the line y = x
putting y = x
x2 + x2 + x2 = 1
3x2 = 1
x = +1/√3 and -1/√3
y = +1/√3 and -1/√3
Thus extremities of one axis of ellipse are (1/√3 , 1/√3 ) and (-1/√3 , -1/√3 )
Now putting y = -x
x2 + x2 - x2 = 1
x = 1 and -1
y = -1 and 1
The extremities of other axis are (-1, 1) and (1, -1)
Thus we found that the major axis of ellipse lies along the line y = -x
Length of largest chord = distance between (-1, 1) and (1, -1)
= √(1 + 1)2 + (-1 - 1)2
= 2√2
(2) Length of chord perpendicular to the major chord and passing through the centre
= distance between (1/√3 , 1/√3 ) and (-1/√3 , -1/√3 )
= √(1/√3 + 1/√3)2 + (-1/√3 - 1/√3)2
= √8/3
= 2√(2/3)
Hence the length of major axis is 2√2 and length of minor axis is 2√(2/3)
SOLUTION:--
First of all the correct equation of ellipse should be x2 + y2 + xy - 1 = 0
The constant 1 appears with positive sign in RHS
We will trace the shape and axes of ellipse
Let f(x,y) = x2 + y2 + xy - 1
(i). We notice that on interchanging x with y, the equation remains unchanged
f(x, y) = f(y, x)
x2 + y2 + xy - 1 = y2 + x2 + yx - 1
Thus the ellipse is symmetric about the line y = x
(ii). Also f(x,y) = f(-y, -x)
The ellipse is symmetric about the line y = -x
From this we conclude that it is an oblique ellipse whose centre is at origin and axes lying along y = x and y = -x
Let us find the coordinates of intersection of ellipse with the line y = x
putting y = x
x2 + x2 + x2 = 1
3x2 = 1
x = +1/√3 and -1/√3
y = +1/√3 and -1/√3
Thus extremities of one axis of ellipse are (1/√3 , 1/√3 ) and (-1/√3 , -1/√3 )
Now putting y = -x
x2 + x2 - x2 = 1
x = 1 and -1
y = -1 and 1
The extremities of other axis are (-1, 1) and (1, -1)
Thus we found that the major axis of ellipse lies along the line y = -x
Length of largest chord = distance between (-1, 1) and (1, -1)
= √(1 + 1)2 + (-1 - 1)2
= 2√2
(2) Length of chord perpendicular to the major chord and passing through the centre
= distance between (1/√3 , 1/√3 ) and (-1/√3 , -1/√3 )
= √(1/√3 + 1/√3)2 + (-1/√3 - 1/√3)2
= √8/3
= 2√(2/3)
Hence the length of major axis is 2√2 and length of minor axis is 2√(2/3)
Answered by
3
First of all the correct equation of ellipse should be x2 + y2 + xy - 1 = 0
The constant 1 appears with positive sign in RHS
We will trace the shape and axes of ellipse
Let f(x,y) = x2 + y2 + xy - 1
(i). We notice that on interchanging x with y, the equation remains unchanged
f(x, y) = f(y, x)
x2 + y2 + xy - 1 = y2 + x2 + yx - 1
Thus the ellipse is symmetric about the line y = x
(ii). Also f(x,y) = f(-y, -x)
The ellipse is symmetric about the line y = -x
From this we conclude that it is an oblique ellipse whose centre is at origin and axes lying along y = x and y = -x
Let us find the coordinates of intersection of ellipse with the line y = x
putting y = x
x2 + x2 + x2 = 1
3x2 = 1
x = +1/√3 and -1/√3
y = +1/√3 and -1/√3
Thus extremities of one axis of ellipse are (1/√3 , 1/√3 ) and (-1/√3 , -1/√3 )
Now putting y = -x
x2 + x2 - x2 = 1
x = 1 and -1
y = -1 and 1
The extremities of other axis are (-1, 1) and (1, -1)
Thus we found that the major axis of ellipse lies along the line y = -x
Length of largest chord = distance between (-1, 1) and (1, -1)
= √(1 + 1)2 + (-1 - 1)2
= 2√2
(2) Length of chord perpendicular to the major chord and passing through the centre
= distance between (1/√3 , 1/√3 ) and (-1/√3 , -1/√3 )
= √(1/√3 + 1/√3)2 + (-1/√3 - 1/√3)2
= √8/3
= 2√(2/3)
Hence the length of major axis is 2√2 and length of minor axis is 2√(2/3)
View Full Answer
The constant 1 appears with positive sign in RHS
We will trace the shape and axes of ellipse
Let f(x,y) = x2 + y2 + xy - 1
(i). We notice that on interchanging x with y, the equation remains unchanged
f(x, y) = f(y, x)
x2 + y2 + xy - 1 = y2 + x2 + yx - 1
Thus the ellipse is symmetric about the line y = x
(ii). Also f(x,y) = f(-y, -x)
The ellipse is symmetric about the line y = -x
From this we conclude that it is an oblique ellipse whose centre is at origin and axes lying along y = x and y = -x
Let us find the coordinates of intersection of ellipse with the line y = x
putting y = x
x2 + x2 + x2 = 1
3x2 = 1
x = +1/√3 and -1/√3
y = +1/√3 and -1/√3
Thus extremities of one axis of ellipse are (1/√3 , 1/√3 ) and (-1/√3 , -1/√3 )
Now putting y = -x
x2 + x2 - x2 = 1
x = 1 and -1
y = -1 and 1
The extremities of other axis are (-1, 1) and (1, -1)
Thus we found that the major axis of ellipse lies along the line y = -x
Length of largest chord = distance between (-1, 1) and (1, -1)
= √(1 + 1)2 + (-1 - 1)2
= 2√2
(2) Length of chord perpendicular to the major chord and passing through the centre
= distance between (1/√3 , 1/√3 ) and (-1/√3 , -1/√3 )
= √(1/√3 + 1/√3)2 + (-1/√3 - 1/√3)2
= √8/3
= 2√(2/3)
Hence the length of major axis is 2√2 and length of minor axis is 2√(2/3)
View Full Answer
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