Let P be the point (-3, 0) and Q be a moving point (0,3t). Let PQ be trisected at R so that R is nearer to Q. RN is drawn perpendicular to PQ meeting the x-axis at N. The locus of the mid-point of RN is

Since R is point of trisection R⇒R divides PQ in the ratio 2:1 =2(0)+1(-3)3, 2(3t)+1(0)3 P(−3,0),Q(0,3t),R(−1,2t) Let the mid point of RN be (h,k)R(-1,2t)  + N(x,y) 2=(h,k)-1+x=2h   2t+y=2kN=(2h+1,0)∴    k=t    RN⊥PQ⇒2t−2h−2×3t3=−1⇒    2t2=2h+2⇒    t2=h+1⇒    k2=h+1 Locus     of (h,k) is y2=x+1