where a2 > b2 and major axis is along x-axis.
Center : (0, 0).
Foci : F(ae, 0) and F'(-ae, 0)
Vertices : A(a, 0) and A'(-a, 0).
Equations of directrices : x = a/e and x = -a/e
Center : (0, 0).
Foci : F(0, ae) and F'(0, -ae)
Vertices : A(0, a) and A'(0, -a).
Equations of directrices : y = a/e and y = -a/e
Center : (0, 0).
Foci : F(ae, 0) and F'(-ae, 0)
Vertices : A(a, 0) and A'(-a, 0).
Equations of directrices : x = a/e and x = -a/e
Center : (0, 0).
Foci : F(0, ae) and F'(0, -ae)
Vertices : A(0, a) and A'(0, -a).
Equations of directrices : y = a/e and y = -a/e
Examples 1-2 : Find center, foci, vertices, and equations of directrices of of the following ellipses :
Example 1 :
Solution :
The given ellipse is symmetric about x-axis.
a2 = 25 a2 = 52 a = 5 |
b2 = 9 b2 = 32 b = 3 |
Center : (0, 0)
Foci :
F(ae, 0) and F'(-ae, 0)
Foci are F(4, 0) and F'(-4, 0).
Vertices :
A(a, 0) and A'(-a, 0)
A(5, 0) and A'(-5, 0)
Equations of directrices :
x = a/e and x = -a/e
Example 2 :
Solution :
The given ellipse is symmetric about y-axis.
a2 = 10 a = √10 |
b2 = 3 a = √3 |
Center : (0, 0)
Foci :
F1 (ae, 0) F2 (-ae, 0)
Foci are F(0, √7) and F'(0, √7).
Vertices :
A(0, a) and A'(0, -a)
A(0, √10) and A'(0, -√10)
Equations of directrices :
y = a/e and y = -a/e
Example 3 :
Solution :
The given hyperbola is symmetric about x-axis.
a2 = 25 a2 = 52 a = 5 |
b2 = 144 b2 = 122 b = 12 |
Center : (0, 0)
Foci :
F(ae, 0) and F'(-ae, 0)
Foci are F(13, 0) and F'(-13, 0).
Vertices :
A (a, 0) A' (-a, 0)
A (5, 0) A' (-5, 0)
Equation of directrices :
x = a/e and x = -a/e
Example 4 :
Solution :
The given hyperbola is symmetric about y-axis.
a2 = 16 a2 = 42 a = 4 |
b2 = 9 b2 = 32 b = 3 |
Center : (0, 0)
Foci :
F(0, ae) and F'(0, -ae)
Foci are F(0, 5) and F'(0, -5).
Vertices :
A(0, a) and A'(0, -a)
A(0, 4) and A'(0, -4)
Equations of directrices :
y = a/e and y = -a/e
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