FINDING INFORMATION FROM EQUATION OF ELLIPSE

Question :

Find the

(i)  Center

(ii)  Foci

(iii)  Eccentricity

(iv)  Equation of latus rectum 

(v)  Equation of directrix

(vi)  Length of latus rectum

1)  9x2+4y2  =  36

2)  25x2+9y2-150x-90y+225  =  0

3)  3x2+4y2-12x-8y+4  =  0

4)  9x2+25y2-18x-100y-116  =  0

1)  Answer :

Equation of ellipse :

9x+ 4y2  =  36

(x2/4) + (y2/9)  =  1

a2  =  9 and b2  =  4

a  =  3 and b  =  2

Since b > a, the ellipse symmetric about y-axis.

e  =  √1 - (4/9)

e  =  √( 5/9)

e  =  √5/3

ae  =  3(√5/3)

ae  =  √5

a/e  =  9/√5

Center : (0, 0)

Focus :  F1 (0, √5) and F2 (0, -√5)

Eccentricity : e  =  √5/3

Equation of latus rectum  :  y  =  ±√5

Equation of directrix :  y  =  ±9/√5

Length of latus rectum :

=  2(4)/3

=  8/3

2)  Answer :

Equation of ellipse :

25x2+9y2-150x-90y+225  =  0

25x2-150x+9y2-90y+225  =  0

25(x2-6x) + 9(y2-10y) + 225  =  0

25[(x-3)2-32] + 9[(y-5)2-52] + 225  =  0

25[(x-3)2-9] + 9[(y-5)2-25] + 225  =  0

25(x-3)2-225 + 9(y-5)2-225 + 225  =  0

25(x-3)+ 9(y-5)2  =  225

(x-3)2/9 + (y-5)2/25  =  1

Let X  =  x-3 and Y  =  y-5

x  =  X + 3 and y  =  Y + 5

The ellipse is symmetric about y-axis.

a2  =  25, b2  =  9

a  =  5 and b  =  3

e  =  √(1-9/25)

e  =  4/5

ae  =  4

a/e  =  25/4

Length of latus rectum :

=  2(9)/5

=  18/5

Referred to

X an Y

Referred to

x an y

x = X+3, y = Y+5

Center

Foci

Vertices

Equation of latus rectum

Equation of directrix

C(0, 0)

F1(0, 4) F2(0, -4)

A(0, 5) A'(0, -5)

Y  =  4

  Y  =  -4

Y  =  25/4

Y  =  -25/4

C(3, -5)

F1(3, 1) F2(3, 9)

A(3, 10) A'(3, 0)

y  =  9

y  =  1

y  =  45/4

y  =  -5/4

3)  Answer :

Equation of ellipse :

3x2+4y2-12x-8y+4  =  0

3x2-12x+4y2-8y+4  =  0

3(x2-4x) + 4(y2-2y) + 4  =  0

3[(x-2)2-22] + 4[(y-1)2-12] + 4  =  0

3[(x-2)2-4] + 4[(y-1)2-1] + 4  =  0

3(x-2)2-12 + 4(y-1)2-4 + 4  =  0

3(x-2)+ 4(y-1)2  =  12

(x-2)2/4 + (y-1)2/3  =  1

Let X  =  x-2 and Y  =  y-1

x  =  X + 2 and y  =  Y + 1

The ellipse is symmetric about x-axis.

a2  =  4, b2  =  3

a  =  2 and b  =  3

e  =  √(1-3/4)

e  =  1/2

ae  =  1

a/e  =  4

Length of latus rectum :

=  2(3)/2

=  3

Referred to

X an Y

Referred to

x an y

Center

Foci

Vertices 

Equation of latus rectum

Equation of directrix

C(0, 0)

F1(1, 0) F2(-1, 0)

A(2, 0) A'(-2, 0)

X  =  1

X  =  -1

X  =  4

X  =  -4

C(2, 1)

F1(3, 1) F2(1, 1)

A(4, 1) A'(0, 1)

x  =  3

x  =  1

x  =  6

x  =  -2

4)  Answer :

Equation of ellipse :

9x2+25y2-18x-100y-116  =  0

9x2-18x+25y2-100y-116 =  0

9(x2-2x) + 25(y2-4y) - 116  =  0

9[(x-1)2-12] + 25[(y-2)2-22] - 116  =  0

9[(x-1)2-1] + 25[(y-2)2-4] - 116  =  0

9(x-1)2-9 + 25(y-2)2-100 -116  =  0

9(x-1)+ 25(y-2)2  =  225

(x-1)2/25 + (y-2)2/9  =  1

Let X  =  x-1 and Y  =  y-2

x  =  X + 1 and y  =  Y + 2

The ellipse is symmetric about x-axis.

a2  =  25, b2  =  9

a  =  5 and b  =  3

e  =  √(1-9/25)

e  =  4/5

ae  =  4

a/e  =  25/4

Length of latus rectum :

=  2(9)/5

=  18/5

Referred to

X an Y

Referred to

x an y

Center

Foci

Vertices

Equation of latus rectum

Equation of directrix

C(0, 0)

F1(4, 0) F2(-4, 0)

F1(5, 0) F2(-5, 0)

X  =  4

X  =  -4

X  =  25/4

X  =  -25/4

C(1, 2)

F1(5, 2) F2(-3, 2)

A'(6, 2) A'(-4, 2)

x  =  5

x  =  -3

x  =  49/4

x  =  -21/4

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