Question

If the roots of \[a(b - c){x^2} + b(c - a)x + c(a - b) = 0\] be equal, then \[a,b,c\] are in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer 1

Hint:
In this question, we are to find the type of progression of the given terms. We will begin by determining the discriminant of the given quadratic equation since it is given that the roots of that equation are zero. So, we can deduce the relationship between \[a,b,c\] from the discriminant. 
Formula used:
We can use the quadratic formula
\[a{x^2} + bx + c = 0\]
The discriminant of the above quadratic equation is calculated by
${{b}^{2}}-4ac=0$ (If roots are equal)
${{b}^{2}}-4ac>0$ (If roots are real)
${{b}^{2}}-4ac<0$ (If roots are imaginary)
Complete Step-by Step Solution:
Given the quadratic equation,
\[a(b - c){x^2} + b(c - a)x + c(a - b) = 0\]
When comparing the above equation to the standard form
\[a{x^2} + bx + c = 0\]
We obtain,
$a=a(b-c);b=b(c-a);c=c(a-b)$
Since the roots are equal, the discriminant of the quadratic equation equals zero.
So, we can write
  \[\ {{b}^{2}}-4ac=0\]
 \[\Rightarrow {{b}^{2}}{{(c-a)}^{2}}-4ac(b-c)(a-b)=0\]
 \[\Rightarrow {{b}^{2}}({{c}^{2}}-2ac+{{a}^{2}})-4ac(ab-{{b}^{2}}-ac+bc)=0\]
 \[ \Rightarrow {{b}^{2}}{{c}^{2}}-2{{b}^{2}}ac+{{a}^{2}}{{b}^{2}}-4{{a}^{2}}bc+4{{b}^{2}}ac+4{{a}^{2}}{{c}^{2}}-4ab{{c}^{2}}=0\]
 \[\Rightarrow {{a}^{2}}{{b}^{2}}+{{b}^{2}}{{c}^{2}}+2{{b}^{2}}ac-4{{a}^{2}}bc-4ab{{c}^{2}}+4{{a}^{2}}{{c}^{2}}=0 \]
 \[ \Rightarrow {{b}^{2}}({{a}^{2}}+2ac+{{b}^{2}})-4ac(ab+bc)+4{{a}^{2}}{{c}^{2}}=0 \]
  \[\Rightarrow {{b}^{2}}{{\left( a+c \right)}^{2}}-4abc(a+c)+4{{a}^{2}}{{c}^{2}}=0\]
 \[\Rightarrow {{\left( b(a+c) \right)}^{2}}-2\left\{ 2ac\left[ b(a+c) \right] \right\}+{{\left( 2ac \right)}^{2}}=0\]
Thus, the obtained expression is in the form of the algebraic formula,
${{(x-y)}^{2}}={{x}^{2}}-2xy+{{y}^{2}}$
On comparing, we get
 \[ \Rightarrow {{\left( b(a+c)-2ac \right)}^{2}}=0\]
 \[\Rightarrow b(a+c)-2ac=0\]
 \[\Rightarrow b=\frac{2ab}{a+c}\]
Thus, the terms a, b, and c are in H.P.
Therefore, if the roots of \[a(b - c){x^2} + b(c - a)x + c(a - b) = 0\] be equal, then \[a,b,c\] are in H.P.
Hence, option C is correct
Note:
Quadratics or quadratic equations are polynomial equations of the second degree. A quadratic equation has the general form \[a{x^2} + bx + c = 0,a \ne 0\]. The expression $D={{b}^{2}}-4ac$ is the discriminant. The student should remember this while solving these types of problems. If \[D > 0\], then the roots are real and different. If \[D < 0\], then the roots of the quadratic equation are imaginary. If \[D = 0\], then the roots of the quadratic equation are equal.