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Question

Prove that the locus of the poles of chords of a parabola which subtend a right angle at a fixed point (h, k) is
ax2hy2+(4a2+2ah)x2aky+a(h2+k2)=0.

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Solution

For y2=4ax let the pole be T(x1,y1) and the point of contact of tangents be P(at21,2at1) and Q(at22,2at2)

Equation of chord of contact with respect to T is

yy1=2a(x+x1)2axyy1+2ax1......(i)

Equation of chord joining PQ is

(t1+t2)y=2x+2at1t22x(t1+t2)y+2at1t2=0.....(ii)

Now (i) and (ii) are the equation of same line

22a=(t1+t2)y1=2at1t22ax1t1+t2=y1a,t1t2=x1a.....(iii)

Given k2at1hat21×k2at2hat22=1

k22ak(t1+t2)+4a2t1t2=(h2ah(t21+t22)+a2t21t22)k22ak(t1+t2)+4a2t1t2={h2ah((t1+t2)22t1t2)+a2t21t22}

Substituting (iii)

k22ak(y1a)+4a2(x1a)={h2ah((y1a)22(x1a))+a2(x1a)2}k22ak(y1a)+4a2(x1a)={h2ah(y212ax1a2)+a2(x1a)2}ak22aky1+4a2x1=(ah2hy21+2hax1+ax21)ax21hy21+(4a2+2ah)x12aky1+a(k2+h2)=0

Generalising the equation

ax2hy2+(4a2+2ah)x2aky+a(k2+h2)=0

Hence proved


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