Let P(x1,y1) be a point on the curve y2=4‌ax ...(i) On differentiating y2=4ax w.r.t. 'x', we get 2y
dy
dx
=4a ⇒ (
dy
dx
)(x1,y1)=
2a
y1
Thus, the equation of normal at (x1,y1) is y−y1=
y1
2a
(x−x1) ⇒ y1x+2‌ay=y1(x1+2a) ...(ii) But lx+my=1 ...(iii) is also a normal. Therefore, coefficients of eqs. (ii) and (iii), must be proportional. i.e.,
y1
l
=
2a
m
=
y1(x1+2a)
1
⇒y1=
2al
m
and x1=
1
l
−2a Putting these values of x1‌and‌y1 in eq. (i), we get (