Let `A-=(-1,0),B-=(3,0),` and `P Q` be any line passing through (4, 1) having slope `mdot` Find the range of `m` for which there exist two points on `P Q` at which `A B` subtends a right angle.

Explanation

The equation of line PQ is
`(y-1)=m(x-4)`
or `y-mx+4m-1=0`
For the required m, we have to make sure that the line PQ meets the circle, with diameter AB, at real and distinct points.
The equation of the circle having AB as diameter is
`x^(2)+y^(2)-2x-3=0`
`:. (|0-m+4m-1|)/(sqrt(1+m^(2)))gt2`
or `5m^(2)-6m-3lt0`
or `m in ((3-2sqrt(6))/(5),(3+2sqrt(6))/(5))`