The line y=3x-3 is tangent to a parabola which has vertex (0,0) and axis of symmetry the y-axis. Find the equation of the parabola.
The general equation of a parabola in Cartesian coordinates is y = ax^2 + bx + c. Since the vertex (h,k) is at (0,0), we know that c = 0 ( plug in x = 0 and y = 0 ).
we also know that since h = -b/4a, b = 0
We also know that somewhere on this parabola, the line y = 3x - 3 is tangent to the parabola. Tangent means that the slopes are the same. Let the line touch the parabola at the point (s,t). So we know the following:
t = 3s - 3
t = a(s^2)
3 = 2a*s ( because the slope of the parabola is y' = 2ax + b )
Here you have 3 equations and 3 unknowns, so solve:
t = 3s/2 = 3s - 3 --> 3s = 6s - 6 --> s = 2
thus, t = 3 and a = 3/4
So, y = 3/4 x^2
kjcdb8er's solution was excellent!
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