If c > 1 / 2, how many lines through the point (0, c) are normal lines to the parabola y = x ^ 2? What if c <= 1/2? Could someone please help?
Let a be the x-value of where the normal intersects x^2. Express m, the gradient of the normal line, in terms of a. Then (at least if you do a sketch) you can see that c, the height of where the normal intersects the y-axis, will be equal to a^2 minus ma, which you should find is equal to a^2 + 1/2. (Which is at least 1/2)
You can have many lines (with different gradients) through a point (0,c) but only 3 of them will hit the curve at right-angles to it (or at right-angles to, rather, the curve's tangent at each of the points where they hit). Which is what it means for a line to be 'normal' to the curve.
But couldnt the curve have many gradients as you arent given the point thats tangent to the curve? meaning the normal wont only cut 1 tangent at 90 degrees but will also cut the other tangents of different slopes? (y=2x gradient seems as if it could cut any where on the parabola)
A line is only a normal if it cuts the curve at 90 degrees, so any cuts it makes with tangents other than at 90 degrees and on the curve are beside the point.
Good that you did the sketch, it will help you - but now make sure you only show lines that are truly (or at least roughly) normal to the curve.
Also, show a, and the geometry and/or algebra that enable you to determine c from it. E.g. the altitude of the point on the curve, a^2, and the height of a triangle with base a and hypotenuse with slope 1/(2a).