Question

The locus of the point $p=(a,b)$ where a,b are real numbers such that the roots of $x^3+a{x}^2+bx+a=0$ are in arithmetic progression is

• A. an ellipse
• B. a circle
• C. a parabola whose vertex is on the y-axis
• D. a parabola whose vertex is on the x-axis

Answer 1

Answer: (C)

a parabola whose vertex is on the y-axis

Let the roots be $\alpha ,\beta ,\gamma$

$\therefore \alpha +\gamma =2\beta$ .....( 1 ) [Given]

$\therefore \alpha +\beta +\gamma =-a$ .....( 2 )

$\therefore \alpha \beta +\alpha \gamma +\beta \gamma =b$ .....( 3 )

$\therefore \alpha \beta \gamma =-a$ .....( 4 )

From eq (1) and (2):

$3\beta =-a$

$\therefore \beta =\frac{-a}{3}$ .....( 5 )

From eq (4) and (5):

$\alpha \gamma =3$ .....( 6 )

From eq (3), (5) and (6):

$(\alpha +\gamma ).\left(\frac{-a}{3}\right)+3=b$

$\Rightarrow 2.\left(\frac{-a}{3}\right).\left(\frac{-a}{3}\right)+3=b$ [From eq (1)]

$\Rightarrow 2a^2+27=9b$

$\therefore a^2=\frac{1}{2}(9b-27)$

$\therefore x^2=\frac{1}{2}(9y-27)$

$\therefore$ Eq of a parabola whose vertex is on the y-axis.