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Answers
Option (c) is correct : 16 : 3
Let α be the one root of the equation and β be the other root .
Given : ax² + bx + c = 0 , α = 3β
On comparing the given equation with ax² + bx + c = 0
Here, a = a , b = b , c = c
Sum of zeroes = - b/a
α + β = - b/a
3β + β = - b/a
4β = - b/a
β = - b/4a ………………(1)
Product of zeroes = c/a
α × β = c/a
3β × β = c/a
3β² = c/a
β² = c/3a ……………..(2)
Put the value of β = - b/4a in eq 2,
(- b/4a)² = c/3a
b²/16a² = c/3a
b² = (c/3a) × 16a²
b² = 16ac/3
b²/ac = 16/3
b² : ac = 16 : 3
Hence, b² : ac is 16 : 3
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Answer:
OPTION B
Step-by-step explanation:
Given one root of the equation ax² + bx + c = 0 is three times the other :
we have to find the ratio of b² : ac
Let us say that one root of the equation is p .
Then the other root of the equation will be 3 p as it is three times the other root .
Hence we have one root p and another root 3 p .
Sum of roots of a quadratic equation = - b / a
Hence : p + 3 p = - b / a
⇒ 4 p = - b / a
⇒ p = - b / 4 a
⇒ p² = b² / 16 a² -------( 1 )
Product of the roots is c / a
Hence p × 3 p = c / a
⇒ 3 p² = c / a
⇒ p² = c / 3 a -----( 2 )
From 1 and 2 we get :-
c / 3 a = b² / 16 a²
⇒ c / 3 = b² / 16 a
⇒ b² / ac = 16 / 3
b² : ac = 16 : 3