Question

The locus of the point P = (a, b) where a, b are real numbers such that the roots of `x^(3)+ax^(2)+bx+a =0` are in arithmetic progression is -
A. an ellipse
B. a circle
C. a parabola whose vertex in on the y-axis
D. a parabola whose vertex is no the x-axis

Answer 1

Correct Answer - C
`{:("Let roots "alpha-d", " alpha", " alpha+d,,"product"),("Sum "3alpha=-a rArr alpha=-a/3,,alpha(alpha^(2)-d^(2))=-a),("pair product "b=alpha^(2)-alphad+alpha^(2)+alphad+alpha^(2)-d^(2),,alpha^(2)-d^(2)=3):}`
`b=2a^(2)+3`
`b-3=2/9 a^(2) rArr" locus "x^(2)=9/2 (y-3)` parabola