Tangent `P Aa n dP B` are drawn from the point `P` on the directrix of the parabola `(x-2)^2+(y-3)^2=((5x-12 y+3)^2)/(160)` . Find the least radius of the circumcircle of triangle `P A Bdot`

Explanation

We have equation of parabola,
`sqrt((x-2)^(2)+(y-3)^(2))=(|5x-12y+3|)/(sqrt(5^(2)+(-12)^(2)))`
Focus of the parabola is (2,3) and directrix is 5x-12y+3=0.
Now, tangents drawn to parabola from point P on the directrix are perpendicular and the corresponding chord of contact AB focal chord which is diameter of the circumcircle of the triangle PAB.
So, least value of diameter is latus rectum.
Here, `L.R=2xx` Distance of focus from directrix
`=2xx(|10-36+3|)/(13)=(23)/(13)`
So, required radius `=(46)/(13)`