All the three vertices of an equilateral triangle lie on the parabola y=x2 and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum equal to pq where p and q are relatively prime positive integers. Find the value of (q - p)

Suppose that t1,t12, t2,t22,t3,t32 are the vertices of an equilateral triangle The slopes of the sides of the triangle are t1+t2,t2+t3,t1+t3Suppose that t1+t2=2Since the angle between any two sides is 600hence, 2−m21+2m2=3  and m3−21+2m3=3 ⇒2-t2-t31+2t2+2t3=3  ⇒2-t2-t3=3+23t2+23t3                   ...... 1t2+t3=2-31+23 -2+t3+t11+2t3+2t1=3 -2+t3+t1=3+23t3+23t1               ......2 t1+t3=2+31-23 t1+t2+2t3=2-31+23+2+31-23 =2-43-3+6+2+3+43+61-12 =-16112t3=-1611-2 =-16-2211t3=-1911t1+t2+t3=-1911+2 =-19+2211 =311Here p=3,q=11, Hence, q-p=8