The locus of the centroid of the triangle with vertices at (a , a ),(b ,-b ) and (1,0) is (here, is a parameter)

Step-by-step solution

Given, vertices of a triangle are A(a , a ), B(b , b ) and C(1,0) Let the locus of centroid be (x, y) . aligned & (x, y)= ( a +b +1 3 , b +0 3 ) & x= a +b +1 3 & and y= a -b 3 aligned and y= a -b 3 a +b =x-1 and a -b =3 y a^2 ^2 +b^2 ^2 +2 a b =(3 x-1)^2 and a^2 ^2 +b^2 ^2 -2 a b =9 y^2 On adding, we get aligned & a^2 ( ^2 + ^2 )+b^2 ( ^2 + ^2 ) &=(3 x-1)^2+9 y^2 & a^2+b^2=(3 x-1)^2+9 y^2 aligned