we can find the point of intersection by equating both equations and verifying that the resulting coordinates match the given point (1, 2).
Let's equate the two equations and solve for x and y:
y^2 = 4x x^2 + y^2 - 6x + 1 = 0
Substituting y^2 = 4x into the second equation:
x^2 + 4x - 6x + 1 = 0
x^2 - 2x + 1 = 0
(x - 1)^2 = 0
Taking the square root of both sides:
x - 1 = 0
x = 1
Now, substitute x = 1 into the first equation to find y:
y^2 = 4(1)
y^2 = 4
y = ±2
So, the points of intersection are (1, 2) and (1, -2).
Since one of the points of intersection is (1, 2), which matches the given point, we can conclude that the curves y^2 = 4x and x^2 + y^2 - 6x + 1 = 0 touch each other at the point (1, 2).
