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Intersection of a Line and a Parabola

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Let the parabola be
y2 = 4ax ...(1)
 
and the given line be
y = mx + c  ...(2)
 
Eliminating y from (1) and (2), then (mx + c)2 = 4ax
 
or m2x2 + 2x(mc – 2a) + c2 = 0 ...(3)
 
This equation, being quadratic in x, gives two values of x, shows that every straight line will cut the parabola in two points may be real, coincident or imaginary accordingly as discriminant of (3) >, =, < 0,
 
i.e., 4(mc – 2a)2 – 4m2c2 >, =, < 0
 
or 4a2 – 4amc >, =, < 0
 
or a >, =, < mc  ...(4)




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