109. if c\ \textgreater \ (1)/(2) , how many lines through the point (0, c) are normal lines to the parabola y=x^(2) ? what if c \leqslant (1)/(2) ?

Question

109. if
`c\ \textgreater \ (1)/(2)` , how many lines through the point
`(0, c)` are normal lines to the parabola
`y=x^(2) ?` what if
`c \leqslant (1)/(2) ?`

Answer 1

A line with slope m passing through the point (0,c) has equation y - c = m(x - 0). For the line to be normal to the parabola y = x2, its derivative must be the negative reciprocal of the derivative of the parabola, which is 2x. So, the slope m of a normal line satisfies:

m = -1/2x = -1/0 = undefined

Since there is no defined slope for a normal line through (0,c), there are infinite normal lines through that point.

When c ≤ 1/2, the point (0,c) lies on or below the parabola. In this case, there are still infinite normal lines through (0,c), but now some of those lines coincide with the parabola itself.