Let two perpendicular chords through A(x1,y1) be PQ and RS. Equation of PQ is ‌
x−x1
cos‌θ
=‌
y−y1
sin‌θ
=r where tan‌θ= slope of PQ Any point on this line may be taken as (x1+r‌cos‌θ,y1+r‌sin‌θ). As the point lies on y2=4ax ∴‌‌(y1+r‌sin‌θ)2=4a(x1+r‌cos‌θ) ⇒r2sin2θ+2y1‌sin‌θ⋅r−4a‌cos‌θ‌r+y12−4ax1=0 ∴‌‌r1r2=‌
−(y12−4ax1)
sin2θ
Similarly r3r4=‌
−(y12−4ax1)
sin2α
where tan‌α= slope of RS Since PQ is perpendicular to RS, ∴‌‌α=90+θ or θ−90 In either case: sin2α=cos2θ Now ‌
