Question

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

Answer 1

Hint: At first, we will find the discriminant of the given quadratic equation.
The nature of the roots of a quadratic equation depends on the discriminant.
From the discriminant, we can find the relationship between \[a,b,c\].
Also, we know that, if \[p,q,r\] are in G.P., then, \[{q^2} = pr\]

Complete step-by-step answer:
It is given that; the roots of \[({a^2} + {b^2}){x^2} - 2b(a + c)x + ({b^2} + {c^2}) = 0\] are equal.
We have to find out the relation between \[a,b,c\].
Let us consider a quadratic equation \[a{x^2} + bx + c = 0,a \ne 0\]. So, the discriminant of this equation is, \[{b^2} - 4ac\].
When the roots of a quadratic equation are equal, then the discriminant is zero.
So, we have, \[{b^2} - 4ac = 0\]
Since, the roots of \[({a^2} + {b^2}){x^2} - 2b(a + c)x + ({b^2} + {c^2}) = 0\] are equal, then its discriminant is also be zero.
Substitute the values in the discriminant we get,
$\Rightarrow$\[4{b^2}{(a + c)^2} = 4({a^2} + {b^2})({b^2} + {c^2})\]
Simplifying we get,
$\Rightarrow$\[{b^2}({a^2} + {c^2} + 2ac) = {a^2}{b^2} + {b^4} + {a^2}{c^2} + {b^2}{c^2}\]
Simplifying again we get,
$\Rightarrow$\[{a^2}{b^2} + {b^2}{c^2} + 2a{b^2}c = {a^2}{b^2} + {b^4} + {a^2}{c^2} + {b^2}{c^2}\]
Rearranging the terms which is equal to 0, we get,
$\Rightarrow$\[{b^4} - 2a{b^2}c + {a^2}{c^2} = 0\]
Simplifying again we get,
$\Rightarrow$\[{({b^2} - 2ac)^2} = 0\]
Simplifying we get,
$\Rightarrow$\[{b^2} = ac\]
We know that, if \[p,q,r\] are in G.P., then, \[{q^2} = pr\]
So, we can say that, \[a,b,c\] are in G.P.

Hence, the correct option is B.

Note: Quadratics or quadratic equations can be defined as a polynomial equation of a second degree. The general form of a quadratic equation is \[a{x^2} + bx + c = 0,a \ne 0\].
The nature of the roots of a quadratic equation depends on the discriminant.
The discriminant is \[D = {b^2} - 4ac\].
If, \[D > 0\], then the roots of the quadratic equation 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 real and equal.