Question

If c > 1/2, how many lines through the point (0; c) are normal to the parabola y = x2?

Answer 1

 a normal line is perpendicular to the curve at the point. f '(x) = 2x, so the slope of the normal line is -1/(2x). Assume that we're dealing with a normal line on the left side of the x-axis. In -x units (x is negative here) it will climb x/2x = 1/2 unit, so the y-intercept of the normal line is x^2 + 1/2. As x approaches 0, the y-intercept approaches 1/2, but does not cross it until x = 0, in which case the normal line is the y-axis. For all the oblique lines, there is a mirror image on the other side, which explains the three lines through (0,c).
so 3 lines are passing through it