Which of the following is/are true about the ellipse `x^2+4y^2-2x-16 y+13=0?` `(a)`the latus rectum of the ellipse is 1. `(b)`The distance between the foci of the ellipse is `4sqrt(3)dot` `(c)`The sum of the focal distances of a point `P(x , y)` on the ellipse is 4. `(d)`Line `y=3` meets the tangents drawn at the vertices of the ellipse at points `P` and `Q` . Then `P Q` subtends a right angle at any of its foci.

Options

A. The latus rectum of the ellipse is 1
B. The distance between the foci of the ellipse is `4sqrt(3)`
C. The sum of the focal distances of a point P(x,y) on the ellipse is 4
D. Liney=3 meets the tangents drawn at the vertices of the ellipse at points P and Q. Then PQ subtends a right angle at any of its foci.

Explanation

`x^(2)+4y^(2)-2x-16y+13=0`
or `(x^(2)-2x+1)+4(y^(2)-4y+4)=4`
or `((x-1)^(2))/(4)+((y-2)^(2))/(1)=1`
`:.` Length of latus rectum `=(2xx1)/(2)=1`
Also, `e=sqrt(1-(1)/(4))=(sqrt(3))/(4)`

or `2ae=2xx2xx(sqrt(3))/(2)=sqrt(3)`
Sum of the focal distance =2a=4
Tenagents at the vertices are
`x-1=+-2`
or `x=3,-1`
Therefore the line y=3 intersects these at points P(3,3) and Q(-1,3)
One of the foci is `S(sqrt(3)+1,2)`
Slop of `PS=(1)/(2-sqrt(3))`
Slopo of `QS=(1)/(-2-sqrt(3))`
`:.` Product of slops `=(1)/(2-sqrt(3))xx(1)/(-2-sqrt(3))=-1`