Find the eccentricity, one of the foci, the directrix, and the length of the latus rectum for the conic `(3x-12)^2+(3y+15)^2=((3x-4y+5)^2)/(25)` .

Answer

e= 1/3; focus `-=` (4,5); directrix : 3x-yy+5=0; L.R.=2/5

Explanation

`(3x-12)^(2)+(3y+15)^(2)=((3x+4y+5)^(2))/(25)`
or `sqrt((x-4)^(2)+(y+5)^(2))=(1)/(3)(|3x-4y+5|)/(5)`
Hence the ratio of distance of he variable point P(x,y) from the foxed point (focus) (4,5) to that from the fixed line (directrix`-=3x-4y+5=0)` is 1/3
Also, the locus is an ellipse and its eccentricity is 1/3
Alos, lenght of the latus reactum
`=2(e)xx` (Distance of (4,5) from the line 3x-4y+5=0
`=2xx(1)/(3)(|3xx4-4xx5+5|)/(5)=(2)/(5)`