Which of the following statement(s) is/are TRUE?

Options

A. Theren are infinite positive integral values of a for which `(13x-1)^(2)+(13y-2)^(2)=((5x+12-1)/(a))^(2)` represents an ellipse
B. The minimum distance of a point (1,2) from the ellipse`4x^(2)+9y^(2)-36y+4=0` is 1.
C. If from a point `P(0,alpha)` two normals other thn the axes are drawn to the ellipse `(x^(2))/(25)+(y^(2))/(16=1` , hen `|alpha|lt9//4`
D. If the length of the latus rectum of the ellipse is one -third of its major axis, then its eccentrticity is equal to `1sqrt(3)`

Explanation

The given equation is
`(x-(1)/(13))^(2)+(y-(2)/(13))^(2)=(1)/(a^(2))=(1)/(a^(2))((5x+12y-1)/(13))^(2)`
It represents and ellipse if
`(1)/(a^(2))lt1 or a^(2)gt 1 or a gt 1`
(2) `4x^(2)+8x+9y^(2)-36y=-4`
or `4(x^(2)+2x+1)+9(y^(2)-4y)+4)=36`

or `((x+1)^(2))/(9)+((y-2)^(2))/(4)=1`
Hence, (-1,2) is center and (1,2) lies on the major axis.
Then the required minimum distance is 1.
The equation of normal at `P(theta)` is `5 sec theta x-4 "cosec" theta y=25-16` , and it passes through `P(o,alpha)`. Therefore,
`alpha=(-9)/(4 "cosec" theta)`
`=(-9)/(4) sin theta`
`or |alpha|lt(9)/(4)`
`(2b^(2))/(a)=(2a)/(3)`
or `3b^(2)=a^(2)(1-e^(2))` , we have `1=3(1-e^(2)) or e=sqrt(2//3)`