Theorem: The condition of tangency states that the line y = mx + c touches the parabola y2 = 4ax at
c = a/m
Proof: let y = mx + c be the line intersecting the parabola y2 = 4ax
y2 = 4ax, y = mx + c
(mx + c)2 = 4ax
m2 x2 + c2 + 2mxc = 4ax
m2 x2 +2(mc – 2a)x + c2 = 0
The line will touch the parabola if it intersects at one point only. This will happen only when the roots are real and coincident.
y = mx + c being a tangent to y2 = 4ax
we must have,
4(mc – 2a)2 – 4m2c2 = 0
4a2 – 4amc = 0
Therefore for all values of m the line y = mx + a/m is always tangent to the parabola y2 = 4ax
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