If largest and smallest value of `(y-4)/(x-3)` is `p` and `q` where `(x,y)` satisfy `x^(2)+y^(2)-2-6y+9=0` then which of the following is true

Options

A. `p+q=4/3`
B. `q=1`
C. `p=4/3`
D. `pq=4/3`

Answer

A, C

Explanation

Given circle is `(x-1)^(2)+(y-3)^(2)=1`
Let of tangent to circle is `y-3=m(x-1)+1sqrt(1+m^(2))`
`(3,4)` lies on axis
`1=2m+sqrt(1+m^(2))`
`(1-2m)^(2)=1+m^(2)`
`4m^(2)-2m+1=1+m^(2)`
`3m^(2)-4m=0`
`m=0, 4/3`
`implies(y-4)/(n-3)=m=0.0r 4/3`
Smallest is 0 and tangent `=4/3`