TP and TQ are tangents to y2=4ax and the normals at p and q meet at a point r on the curve. find the locus of the center of the triangle tpq.

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To solve this question, your should have complete theoretical knowledgeabout tangent and normals of parabola because various results will be useddirectly.We have y2=4axLet co-ordinates of P and Q be at12,-2at1 and at22,-2at2 respectively.We know that point of intersection of tangents TPand TQ isat1t2, at1+t2Now equation of normal at P will be:y=-t1x+2at1+at13Let this normal pass through R at32,-2at3 on the parabola. Then we know thatt1+2t1=-t3 ;iSimilarly if normal at Qpasses through R, then:t2+2t2=-t3 ;iiBy i and iit1+2t1=t2+2t2t1-t2+21t1-1t2=0t1-t2-2t1t2t1-t2=0t1-t21-2t1t2=0t1-t20 as P and Qcannot be same.1-2t1t2=0t1t2=2We want centre of triangle i.e. circumcentre.Let the centre of the circle passing through P,Q and T be x,y, hence we h2x=at1+t22+2a ;iii2y=at1+t21-t1t22y=at1+t21-2 As t1t2=22y=-at1+t2t1+t2=-2yaPut this in iii2x=a-2ya2+2a2x-2a=a4y2a24y2a=2x-a2y2=ax-a

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