Tangents drawn from the point `P(2,3)` to the circle `x^2 + y^2-8x + 6y + 1 = 0` points A and B. The circumcircle of the `DeltaPAB` cuts the director circle of ellipse `(x-5)^2/9+(y-3)^2/b^2=1` orthogonally. Find the value of `b^2`.

Explanation

The center of the given circle is O(4,3)

The crircumericle of `DeltaPAB` will circumscribe the quadrilateral PBOA alos. Hence one of the diameters must be OP
So, the quations of circumcricle of `DeltaPAB` will be
`(x-2)(x-4)+(y-3)(y+3)=0`
or `x^(2)+y^(2)-6x=0" "(1)`
Director circle of the given ellipe will be
`(x+5)^(2)+(y-3)^(2)=9+b^(2)`
or `x^(2)+y^(2)+10x-6y+25=0 " "(2)`
So,from (1) and (2), by applying the condition of orthogonoality we ge
`2[-3(5)+0(-3)]=-1+25-b^(2) or -30=24-b^(2)`
Hence, `b^(2)=54`