the line lx + my +n=0 is a normal to the parabola y^2 = 4ax if

the equation of the given parabola is y2=4ax 
the equation of the normal to the given parabola is y+tx=2at+at3 
i.e. tx+y-(2at+at3)=0 ........(1)
 the equation (1) and  the given equation of the normal lx+my+n=0 ...........(2) represent the same line:
tl=1mt=lm ........(3)and 1m=-(2at+at3)n-nm=2at+at3-nm=2a.lm+a.lm3   [ substituting the value of t from eq(3)]-nm=2a.lm+a.l3m3-n.m2=2a.lm2+a.l3a.l3+2alm2+nm2=0

hope this helps you

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