Tangents are drawn from A(x1,y1) and let the point of contact be P(at21,2at1),Q(at22,2at2)
PQ=√(at22−at21)2+(2at2−2at1)2PQ=a√(t2−t1)2(t2+t1)2+4(t2−t1)2PQ=a(t2−t1)√(t2+t1)2+4...........(i)
Equation of chord of contact w.r.t. A is T=0
yy1=2a(x+x1)2ax−yy1+2ax1=0......(ii)
Equation of chord joining AB is
(t1+t2)y=2x+2at1t22x−(t1+t2)y+2at1t2=0......(iii)
Comparing (ii) and (iii)
22a=−(t1+t2)−y1=2at1t22ax11a=t1+t2y1=t1t2x1⇒t1+t2=y1a,t1t2=x1a(t1+t2)2=y21a2...........(iv)(t1−t2)2=(t1+t2)2−4t1t2(t1−t2)2=y21a2−4x1a=y21−4ax1a2t1−t2=√y21−4ax1a.........(v)
substituting (iv) and (v) in (i)
PQ=a√y21−4ax1a√y21a2+4PQ=a√y21−4ax1a√y21+4a2aPQ=√y21−4ax1√y21+4a2a