The length of the axes of the conic `9x^2 + 4y^2 - 6x + 4y +1=0`, are

Options

A. `(1)/(2),9`
B. `3,(2)/(5)`
C. `1,(2)/(3)`
D. `3,2`

Answer

C

Explanation

the equation of the conic is
`9x^(2)+4y^(2)-6x+4y+1=0`
`implies {(3x)^(2)-6x+1}+{(2y)^(2)+4y +1}=1`
` implies (3x-1)^(2)+(2y+1)^(2)=1`
`implies 9(x-(1)/(3))^(2)+4(y+(1)/(2))^(2)=1implies {:((x-(1)/(3))^(2),(y+(1)/(2))^(2)),(((1)/(3))^(2),((1)/(2))^(2)):}=1`
this represents an ellipse whose major axes are of lengths `2xx(1)/(2)=1 and 2xx(1)/(3)=(2)/(3)` respectively .