Math CentralQuandaries & Queries


Question from kanchan, a student:

for what value of c a line y=mx+c touches a parabola y^2=4a(x-a)


If the line y = mx + c just touches the parabola at one point then the line is tangent to the parabola at that point. Let (p,q) be this point then the slope of the tangent at that point is the slope of the line which is m. But, using calculus, the slope of the tangent to the parabola at (p,q) is y' where 2q y' = 4a. Thus

2q m = 4a.

Also (p,q) is on the parabola so

q2 = 4a(p - a).

Furthermore (p,q) is on the line so it satisfies the equation of the line. So altogether you have three equations. Solve these equations for c in terms of a and m.


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