Let P and Q be distinct points on the parabola y2=2x such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle ΔOPQ is 32, then which of the following is (are) the coordinates of P?

LetPat12,2at1Qat22,2at2 be any two points on the parabola y2=2x⇒4a=2 a=12Now the diametric form of equation of the circle is given byx-at12x-at22+y-2at1y-2at2=0Since the circle passes through  (0,0),⇒a2t12t22+4a2t1t2=0⇒a2t1t2t1t2+4=0⇒t2=-4t1    →1 ↔given area ofΔOPQ=32⇒12at12×2at2-2at1at22=32⇒2a2t1t22t1-t2=32⇒t1-t2=32t1-4t1=32   by 1⇒t1-4t1=32,  t1-4t1=-32⇒t12-32t1-4=0, t12+32t1-4=0⇒t12-22t1-2t1-4=0⇒t1t1-22-2t1-22=0⇒t1=2( or ) 22⇒Pat12,2at1=(1,2)(or)(4,22)