where a2 > b2 and major axis is along x-axis.
Center : (0, 0).
Vertices : A(a, 0) and A'(-a, 0).
Co-vertices : B(0, b) and B(0, -b).
Foci : F(ae, 0) and F'(-ae, 0)
where a2 > b2 and major axis is along y-axis.
Center : (0, 0).
Vertices : A(0, a) and A'(0, -a).
Co-vertices : B(b, 0) and B(-b, 0).
Foci : F(o, ae) and F'(0, -ae)
Find the center, vertices and co-vertices of the following ellipses.
Example 1 :
Solution :
The above ellipse is symmetric about x-axis.
a2 = 25 a2 = 52 a = 5 |
b2 = 9 b2 = 32 b = 3 |
Center : (0, 0).
Vertices :
A(a, 0) and A'(-a, 0)
A(5, 0) and A'(-5, 0)
Co-vertices :
B(0, b) and B'(0, -b)
B(0, 3) and B'(0, -3)
Example 2 :
Let X = x - 1 and Y = y + 1.
The above ellipse is symmetric about Y-axis.
a2 = 16 a2 = 42 a = 4 |
b2 = 9 b2 = 32 b = 3 |
Center :
(0, 0)
X = 0 and Y = 0
Substitute X = x - 1 and Y = y + 1.
x - 1 = 0 and y + 1 = 0
x = 1 and y = -1
The center is (1, -1)
Vertices :
A(0, a) and A'(0, -a)
A(0, 4) and A'(0, -4)
(0, 4) X = 0 and Y = 4 x - 1 = 0 and y + 1 = 4 x = 1 and y = 3 (1, 3) |
(0, -4) X = 0 and Y = -4 x - 1 = 0 and y + 1 = -4 x = 1 and y = -5 (1, -5) |
The vertices are (1, 3) and (1, -5).
Co-vertices :
B(b, 0) and B'(-b, 0)
B(3, 0) and B'(-3, 0)
(3, 0) X = 3 and Y = 0 x - 1 = 3 and y + 1 = 0 x = 4 and y = -1 (4, -1) |
(-3, 0) X = -3 and Y = 0 x - 1 = -3 and y + 1 = 0 x = -2 and y = -1 (-2, -1) |
The co-vertices are (4, -1) and (-2, -1).
Example 3 :
4x2 + 32x + 36y2 - 72y - 44 = 0
Solution :
The given equation of ellipse is not in standard form. Convert it to standard form.
4x2 + 32x + 36y2 - 72y - 44 = 0
4(x2 + 8x) + 36(y2 - 2y) - 44 = 0
4[x2+ 2x(4) + 42 - 42] + 36[y2 - 2y(1) + 12 - 12] - 44 = 0
4[(x + 4)2- 16] + 36[(y - 1)2- 1] - 44 = 0
4(x + 4)2- 64 + 36(y - 1)2- 36 - 44 = 0
4(x + 4)2 + 36(y - 1)2 - 144 = 0
4(x + 4)2 + 36(y - 1)2 = 144
Divide both sides by 144.
The above ellipse is symmetric about X-axis.
a2 = 36 a2 = 62 a = 6 |
b2 = 4 b2 = 22 b = 2 |
Center :
(0, 0)
X = 0 and Y = 0
Substitute X = x + 4 and Y = y - 1.
x + 4 = 0 and y - 1 = 0
x = -4 and y = 1
The center is (-4, 1)
Vertices :
A(a, 0) and A'(-a, 0)
A(6, 0) and A'(-6, 0)
(6, 0) X = 6 and Y = 0 x + 4 = 6 and y - 1 = 0 x = 2 and y = 1 (2, 1) |
(-6, 0) X = -6 and Y = 0 x + 4 = -6 and y - 1 = 0 x = -10 and y = 1 (-10, 1) |
The vertices are (2, 1) and (-10, 1).
Co-vertices :
B(0, b) and B'(0, -b)
B(0, 2) and B'(0, -2)
(0, 2) X = 0 and Y = 2 x + 4 = 0 and y - 1 = 2 x = -4 and y = 3 (-4, 3) |
(0, -2) X = 0 and Y = -2 x + 4 = 0 and y - 1 = -2 x = -4 and y = -1 (-4, -1) |
The co-vertices are (-4, 3) and (-4, -1).
Example 4 :
(x + 3)2 / 6 + (y - 5)2 / 4 = 1
Solution :
(x + 3)2 / 6 + (y - 5)2 / 4 = 1
Let X = x + 3 and Y = y - 5
The ellipse is symmetric about X-axis.
x + 3 = 0 and y - 5 = 0
x = -3 and y = 5
The center is (-3, 5)
Vertices :
a2 = 6 and b2 = 4
a = √6 and b = 2
A(a, 0) and A'(-a, 0)
Vertices of major axis referred to X and Y :
A(√6, 0) and A'(-√6, 0)
x + 3 = √6 x = √6 - 3 |
x + 3 = -√6 x = -√6 - 3 |
Vertices of major axis referred to x and y :
A (√6 - 3, 5) A' (-√6 - 3, 5)
Co-vertices :
B(0, b) and B'(0, -b)
Vertices of minor axis referred to X and Y :
B(0, 2) and B'(0, -2)
y - 5 = 2 x = 2 + 5 x = 7 |
y - 5 = -2 x = -2 + 5 x = 3 |
Vertices of minor axis referred to x and y :
B(-3, 7) and B'(-3, 3)
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