Question

19.Centre at (0,0), major axis on the y-axis and passes through the points (3, 2) and

Answer 1

It is given that the center is at ( 0,0 ) , major axis is on y axis and the ellipse passes through (3,2) and ( 1,6 ) .

Since y axis is the major axis, so the equation of the ellipse is represented as x 2 b 2 + y 2 a 2 =1 , where y is the major axis.(1)

( 3,2 ) and ( 1,6 ) lie on the ellipse. Therefore, they satisfy equation ( 1 )

Substitute ( 3,2 ) in equation ( 1 )

3 2 b 2 + 2 2 a 2 =1 9 b 2 + 4 a 2 =1 (2)

Substitute ( 1,6 ) in equation ( 1 )

1 2 b 2 + 6 2 a 2 =1 9( 1 b 2 + 36 a 2 =1 ) 9 b 2 + 324 a 2 =1 (3)

Subtract equation (2) from equation (3),

4 a 2 − 324 a 2 =−8 320 8 = a 2 a 2 =40

Substitute a 2 =40 in (2) to determine b 2 .

9 b 2 + 4 40 =1 9 b 2 =1− 4 40 9 b 2 = 40−4 40 360 36 = b 2

Hence, b 2 =10

Substituting the values of a 2 and b 2 in equation (1), we get

x 2 10 + y 2 40 =1 .

Thus, the equation of the ellipse with the center at ( 0,0 ) , major axis on the y axis and the ellipse passing through (3,2) and ( 1,6 ) is x 2 10 + y 2 40 =1 .