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

Answer

Proved

Explanation

For first curve `2 y(\frac{d y^{x}}{d x})_{1}=4` or `(\frac{d y}{d x})_{1}=\frac{2}{y}`


For second curve `2 x+2 y(\frac{d y}{d x})_{2}=6`


or `(\frac{d y}{d x})_{2}=\frac{3-x}{y}`


As both curves intersect each other at `(1,2)` this means curves are passing through points `(1,2)`


`\therefore` Now at point slope would be `(1,2),(\frac{d y}{d x})_{1}=\frac{2}{2}=1`

and `(\frac{d y}{d x})_{2}=\frac{3-1}{2}=1`


As we can compare the slope of both the curves at point `(1,2)` are equal. This implies both the curves intersect each other at this point.