Prove that the curves `y^2=4x` and `x^2+y^2-6x+1=0` touch each other at the points `(1,\ 2)` .

Explanation

We have, `y^(2)=4x` and `x^(2)+y^(2)-6x+1=0`
Since, both the curves touch each other at (1,2) i.e., curves are passing through (1,2).
`therefore 2y.(dy)/(dx)=4`
and `(dy)/(dx) = 4/(2y)`
and `(dy)/(dx)= (6-2x)/(2y)`
`rArr (dy)/(dx)_(1.2) = 4/4=1`
and `(dy)/(dx)_(1/2) = ((6-2).1)/(2.2) = 4/4=1`
`rArr m_(1)=1` and `m_(2)=1`
Thus, we see that slope of both the curves are equal to each other i.e, `m_(1)=m_(2)=1` at the point (1.2).
Hence, both the curves touch each other.