Symmetric about x- axis Center : C (0, 0) Foci : F1 (ae, 0) F2 (-ae, 0) Vertices on major axis : A (a, 0) A' (-a, 0) Vertices on minor axis : B (0, b) B' (0, -b) Equation of directrices : x = ± (a/e) |
Symmetric about y- axis Center : C (0, 0) Foci : F1 (0, ae) F2 (0, -ae) Vertices on major axis : A (0, a) A' (0, -a) Vertices on minor axis : B (b, 0) B' (-b, 0) Equation of directrices : y = ± (a/e) |
Symmetric about x- axis Center : C (h, k) Foci : F1 (h − c, k ) F2 (h + c, k ) Vertices on major axis : A (h -a, k) A' (h + a, k) Equation of directrices : x = h ± (a/e) |
Symmetric about y- axis Center : C (h, k) Foci : F1 (h, k - c) F2 (h, k + c) Vertices on major axis : A (h, k - a ) A' (h, k + a) Equation of directrices : y = k ± (a/e) |
Note :
e = √[1 + (b2/a2)]
b2 = a2(e2 - 1)
Question 1 :
Identify the type of conic and find centre, foci, vertices, and directrices of the following :
[(x - 3)2/225] + [(y - 4)2/289] = 1
Solution :
The given conic represents the " Ellipse "
The given ellipse is symmetric about y - axis.
a2 = 289 and b2 = 225
a = 17 and b = 15
c2 = a2 - b2
c2 = 289 - 225
c2 = 64
c = 8
e = c/a = 8/17
Center :
C (h, k) ==> (3, 4)
Foci :
F1 (h, k - c ) F2 (h, k + c )
F1 (3, 4 - 8 ) F2 (3, 8 + 4 )
F1 (3, -4 ) F2 (3, 12 )
Vertices :
A (h, k - a ) A' (h, k + a)
A (3, 4 - 17 ) A' (3, 4 + 17)
A (3, -13 ) A' (3, 21)
Equation of directrices :
y = k ± (a/e)
y = 4 ± (17/(8/17))
y = 4 ± (289/8)
y = 4 + (289/8) y = (32+289)/8 y = 321/8 |
y = 4 - (289/8) y = (32 - 289)/8 y = - 257/8 |
Question 1 :
Identify the type of conic and find centre, foci, vertices, and directrices of the following :
[(x + 1)2/100] + [(y - 2)2/64] = 1
Solution :
The given conic represents the " Ellipse "
The given ellipse is symmetric about x - axis.
a2 = 100 and b2 = 64
a = 10 and b = 8
c2 = a2 - b2
c2 = 100 - 64
c2 = 36
c = 6
e = c/a = 6/10 = 3/5
Center :
C (h, k) ==> (-1, 2)
Foci :
F1 (h - c, k) F2 (h + c, k)
F1 (-1-6, 2) F2 (-1 + 6, 2)
F1 (-7, 2) F2 (5, 2)
Vertices :
A (h - a, k) A' (h + a, k)
A (-1 - 10, 2) A' (-1 + 10, 2)
A (-11, 2) A' (9, 2)
Equation of directrices :
x = h ± (a/e)
x = -1 ± (10/(3/5))
x = -1 ± (50/3)
x = -1 + (50/3) x = (-3+50)/3 x = 47/3 |
x = -1 - (50/3) x = (-3-50)/3 x = -53/3 |
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