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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...

If the roots of (a2+b2)x2−2b(a+c)x+(b2+c2)=0({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

Explanation

Solution

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,ca,b,c.
Also, we know that, if p,q,rp,q,r are in G.P., then, q2=pr{q^2} = pr

Complete step-by-step answer:
It is given that; the roots of (a2+b2)x2−2b(a+c)x+(b2+c2)=0({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,ca,b,c.
Let us consider a quadratic equation ax2+bx+c=0,a≠0a{x^2} + bx + c = 0,a \ne 0. So, the discriminant of this equation is, b2−4ac{b^2} - 4ac.
When the roots of a quadratic equation are equal, then the discriminant is zero.
So, we have, b2−4ac=0{b^2} - 4ac = 0
Since, the roots of (a2+b2)x2−2b(a+c)x+(b2+c2)=0({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$$${b^2} = acWeknowthat,if We know that, ifp,q,rareinG.P.,then,are in G.P., then,{q^2} = prSo,wecansaythat, So, we can say that,a,b,c$$ 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 ax2+bx+c=0,a≠0a{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=b2−4acD = {b^2} - 4ac.
If, D>0D > 0, then the roots of the quadratic equation are real and different.
If, D<0D < 0, then the roots of the quadratic equation are imaginary.
If, D=0D = 0, then the roots of the quadratic equation are real and equal.