What's the equation of tangent to the parabola y2=4ax having a slope 'm'.
y=mx+am
Given parabola is y2=4ax .........(1)
Let the tangent of the parabola be,
y=mx+c ..............(2)
for finding the tangent we solve the equations
of line and parabola and take the case when the number
of solution is equal to 1.
(mx+c)2=4ax
m2x2+2mcx+c2=4ax
m2x2+x(2mc−4a)+c2=0
for having a single solution
△=0
⇒(2mc−4a)2−4m2c2=0
⇒4m2c2+16a2−16mca−4m2c2=0
⇒16a2=16mca
⇒a=mc
⇒c=am
Putting the value of c in (c) we get the equation of tangent as,
y=mx+am
From solving the line with the parabola we also get the point of contacts as (am2,2am)