The locus of the point P=(a, b) where a, b ar | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the roots of the equation $x^3+a x^2+b x+a=0$ be $\alpha - d, \alpha, \alpha + d$.From the properties of roots,the sum of roots is $(\alpha -

Vedclass · Accepted Answer

a parabola whose vertex is on the $Y$-axis

Vedclass · Answer

(C) Let the roots of the equation $x^3+a x^2+b x+a=0$ be $\alpha - d, \alpha, \alpha + d$.From the properties of roots,the sum of roots is $(\alpha - d) + \alpha + (\alpha + d) = 3\alpha = -a$,so $\alpha = -a/3$.Since $\alpha$ is a root,it must satisfy the equation: $(-a/3)^3 + a(-a/3)^2 + b(-a/3) + a = 0$.$-a^3/27 + a^3/9 - ab/3 + a = 0$.Multiplying by $27$,we get $-a^3 + 3a^3 - 9ab + 27a = 0$,which simplifies to $2a^3 - 9ab + 27a = 0$.Since $a 
eq 0$ (if $a=0$,the equation becomes $x^3+bx=0$,roots are $0, \pm \sqrt{-b}$,which are in $AP$ only if $b=0$,but $a=0, b=0$ is a point),we divide by $a$: $2a^2 - 9b + 27 = 0$.Thus,$2a^2 = 9b - 27$,or $b = \frac{2}{9}a^2 + 3$.Replacing $(a, b)$ with $(x, y)$,the locus is $y = \frac{2}{9}x^2 + 3$,which is a parabola with its vertex on the $Y$-axis at $(0, 3)$.

$A. \ 2x-5=0$	$I. \ \text{Vertex}$
$B. \ (\frac{3}{2}, -1)$	$II. \ \text{Focus}$
$C. \ y+1=0$	$III. \ \text{Equation of directrix}$
$D. \ (2, -1)$	$IV. \ \text{Equation of the axis}$
	$V. \ \text{Equation of the Latus rectum}$

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

Similar Questions

The parabolas $ax^2 + 2bx + cy = 0$ and $dx^2 + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in $G.P.$,then

Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$,focus $(-2, 0)$.

Find the area of the triangle formed by the lines joining the vertex of the parabola $x^{2}=12y$ to the ends of its latus rectum. (in $\text{ unit}^{2}$)

Find the point of contact of the line $2x - y + 2 = 0$ with the parabola $y^2 = 16x$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module