IDENTIFYING THE TYPE OF CONIC AND FIND FOCI VERTICES AND DIRECTRICES

Note :

e  =  √[1 + (b²/a²)]

b²  =  a²(e² - 1)

Solved Questions

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)

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)

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