Find the equation of the ellipse in the form ( x - h ) 2 a 2 + ( y - k ) 2 b 2 = 1 , given center ( 2 , - 1 ) , one end of the major axis ( 2 , - 5 ) and e = 1 3 .

Given centre of the ellipse h , k = 2 , - 1 and major axis A ( h , k ± b ) = 2 , - 5 and e = 1 3 ( x - h ) 2 a 2 + ( y - k ) 2 b 2 = 1 , ( a < b ) Therefore, we have h = 2 , k = - 1 and b = - 4 We have e 2 = 1 - a 2 b 2 ⇒ a 2 16 = 1 - 1 9 ⇒ a 2 = 128 9 Hence, the equation is ( x - 2 ) 2 128 9 + ( y + 1 ) 2 16 = 1 ⇒ 9 ( x - 2 ) 2 128 + ( y + 1 ) 2 16 = 1

Find the equation of the ellipse in the form ( x - h ) 2 a 2 + ( y - k ) 2 b 2 = 1 , given center ( 4 , - 1 ) , one end of the major axis ( - 1 , - 1 ) and passes through 8 , 0

Given centre of the ellipse h , k = 4 , - 1 and one end of major axis is A = - 1 , - 1 We can observe, y - coordinates of above points are same, major axis is parallel to x -axis a = C A = ( 4 + 1 ) 2 + ( - 1 + 1 ) 2 = 5 Ellipse is passing through 8 , 0 ⇒ ( 8 - 4 ) 2 25 + ( 0 + 1 ) 2 b 2 = 1 ⇒ 1 b 2 = 1 - 16 25 = 9 12 Equation of ellipse is ( x - 4 ) 2 25 + 9 12 ( y + 1 ) 2 = 1

Find the equation of the ellipse in the form ( x - h ) 2 a 2 + ( y - k ) 2 b 2 = 1 , given centre ( 0 , - 3 ) , e = 2 3 , semi minor axis 5 .

Given centre of the ellipse h , k = 0 , - 3 , e = 2 3 and semi minor axis b = 5 ⇒ b 2 = a 2 - a 2 e 2 ⇒ 25 = a 2 - a 2 4 9 = a 2 5 9 ⇒ 45 = a 2 Therefore, equation of ellipse is ( x - 0 ) 2 45 + ( y + 3 ) 2 25 = 1 ⇒ x 2 45 + ( y + 3 ) 2 25 = 1

Find the equation of the ellipse in the form ( x - h ) 2 a 2 + ( y - k ) 2 b 2 = 1 , given center 2 , - 1 , e = 1 2 and length of latus rectum 4

Given centre of the ellipse h , k = 2 , - 1 , e = 1 2 and latus rectum 2 b 2 a = 4 ⇒ b 2 = 2 a ⇒ b 2 = a 2 - a 2 e 2 ⇒ b 2 = a 2 - a 2 1 4 ⇒ b 2 = 3 4 a 2 ⇒ 2 a = 3 4 a 2 ⇒ 8 3 = a or a 2 = 64 9 and b 2 = 16 3 Therefore, equation of ellipse is 9 ( x - 2 ) 2 64 + 3 ( y + 1 ) 2 16 = 1