Step 1: Use Vieta's Formulas
For a cubic equation of the form Ax3+Bx2+Cx+D=0 with roots a,b, and c, the relations between roots and coefficients are:
Given the equation x3−3px2+3qx−1=0:
-
a+b+c=−(1−3p)=3p
-
ab+bc+ca=13q=3q
-
abc=−(1−1)=1
Step 2: Define the Centroid formula
The centroid G(x,y) of a triangle with vertices (x1,y1),(x2,y2), and (x3,y3) is given by:
G=(3x1+x2+x3,3y1+y2+y3)
Step 3: Calculate the x-coordinate of the centroid
Using the sum of roots from Step 1:
xcentroid=3a+b+c=33p=p
Step 4: Calculate the y-coordinate of the centroid
The y-coordinates are the reciprocals of the roots:
To simplify the numerator, find a common denominator:
Substitute the values from Vieta's formulas:
Now, plug this back into the y-coordinate formula:
Conclusion:
The centroid of the triangle is (p,q).
Correct Option: (a)
