Q. Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation .
1 . x² + 18x + 16 = 0
2 . x² - 18x - 16 = 0
3 . x² + 18x - 16 = 0
4 . x² - 18x + 16 = 0
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Q. The equation whose roots are the Arithmetic mean and twice the H.M between the roots of the equation x² - ax - b = 0 is
1. 2ax² + ( a² - 8b )x + 4ab = 0
2. 2ax² + ( a² - 8b )x - 4ab = 0
3. 2ax² + ( a² + 8b )x - 4ab = 0
4. None
Note : -
Irrelevant answers will be reported.
Answers
Question (1)
Given,
▶ Arithmetic mean, A.M = 9
▶ Geometric mean, G.M = 4
We have to find the quadratic equation that has its roots as the two numbers whose A.M and G.M are 9 & 4 respectively.
Let the numbers be, a and b.
According to the Question,
⇒ Arithmetic mean = 9
⇒ (a + b) / 2 = 9
⇒ a + b = 18 ...(1)
Also,
⇒ Geometric mean = 4
⇒ √ab = 4
Square both sides,
⇒ ab = 16 ...(2)
(1) can be called as sum of roots and (2) product of roots. And we know, A quadratic equation can be represented as,
⇒ x² - (sum of roots)x + (product of roots) = 0
⇒ x² - 18x + 16 = 0
Hence, The required quadratic equation is x² - 18x + 16 = 0
So, Option (4) is correct.
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Question (2)
Given,
▶ A quadratic equation, x² - ax - b = 0
▶ Whose roots are such that their arithmetic mean and twice of Harmonic mean are the roots of another quadratic equation.
we have to find that quadratic equation.
From the quadratic equation, x² - ax - b = 0,
⇒ Sum of roots = -(coefficient of x) / (coefficient of x²)
⇒ Sum of roots = -(-a) / 1
⇒ Sum of roots = a
Divide both sides by 2 such that the left side is the arithmetic mean,
⇒ A.M = a/2 ...(1)
Now,
Let's find the twice of Harmonic mean of the roots of the quadratic equation x² - ax - b = 0,
⇒ 2H.M = 2 × 2 / (1/p + 1/q)
Where, p & q are roots.
⇒ 2H.M = 4 /{ ( p + q)/pq }
Where,
▶ (p + q) = sum of roots
▶ pq = product of roots
So,
⇒ H.M = 4pq / (p + q)
⇒ 2H.M = 4(-b) / (a
⇒ 2H.M = -4b/a ...(2)
The arithmetic mean (1) and twice of Harmonic mean (2) are the roots of a quadratic equation and we have to find that quadratic equation.
First, We would have to find the sum and product of roots for getting the required quadratic equation.
⇒ Sum of roots = (1) + (2)
⇒ Sum of roots = a/2 + (-4b/a)
⇒ Sum of roots = a/2 - 4b/a
⇒ Sum of roots = (a² - 8b)/2a ...(3)
In the same way,
⇒ Product of roots = (1) × (2)
⇒ Product of roots = a/2 × -4b/a
⇒ Product of roots = -2b ...(4)
As we know, A quadratic equation can be expressed in the form,
⇒ x² - (sum of roots)x + (product of roots) = 0
⇒ x² - { (a² - 8b) / 2a }x + (-2b) = 0
⇒ x² - { (a² - 8b) / 2a }x - 2b = 0
Multiply both sides by 2a
⇒ 2ax² - (a² - 8b)x - 4ab = 0
∴ The required quadratic equation is
2ax² - (a² - 8b)x - 4ab = 0
So, Option (D) - None, is correct.
Answer:
Question (1)
Given,
▶ Arithmetic mean, A.M = 9
▶ Geometric mean, G.M = 4
We have to find the quadratic equation that has its roots as the two numbers whose A.M and G.M are 9 & 4 respectively.
Let the numbers be, a and b.
According to the Question,
⇒ Arithmetic mean = 9
⇒ (a + b) / 2 = 9
⇒ a + b = 18 ...(1)
Also,
⇒ Geometric mean = 4
⇒ √ab = 4
Square both sides,
⇒ ab = 16 ...(2)
(1) can be called as sum of roots and (2) product of roots. And we know, A quadratic equation can be represented as,
⇒ x² - (sum of roots)x + (product of roots) = 0
⇒ x² - 18x + 16 = 0
Hence, The required quadratic equation is x² - 18x + 16 = 0
So, Option (4) is correct.
___________________________
Question (2)
Given,
▶ A quadratic equation, x² - ax - b = 0
▶ Whose roots are such that their arithmetic mean and twice of Harmonic mean are the roots of another quadratic equation.
we have to find that quadratic equation.
From the quadratic equation, x² - ax - b = 0,
⇒ Sum of roots = -(coefficient of x) / (coefficient of x²)
⇒ Sum of roots = -(-a) / 1
⇒ Sum of roots = a
Divide both sides by 2 such that the left side is the arithmetic mean,
⇒ A.M = a/2 ...(1)
Now,
Let's find the twice of Harmonic mean of the roots of the quadratic equation x² - ax - b = 0,
⇒ 2H.M = 2 × 2 / (1/p + 1/q)
Where, p & q are roots.
⇒ 2H.M = 4 /{ ( p + q)/pq }
Where,
▶ (p + q) = sum of roots
▶ pq = product of roots
So,
⇒ H.M = 4pq / (p + q)
⇒ 2H.M = 4(-b) / (a
⇒ 2H.M = -4b/a ...(2)
The arithmetic mean (1) and twice of Harmonic mean (2) are the roots of a quadratic equation and we have to find that quadratic equation.
First, We would have to find the sum and product of roots for getting the required quadratic equation.
⇒ Sum of roots = (1) + (2)
⇒ Sum of roots = a/2 + (-4b/a)
⇒ Sum of roots = a/2 - 4b/a
⇒ Sum of roots = (a² - 8b)/2a ...(3)
In the same way,
⇒ Product of roots = (1) × (2)
⇒ Product of roots = a/2 × -4b/a
⇒ Product of roots = -2b ...(4)
As we know, A quadratic equation can be expressed in the form,
⇒ x² - (sum of roots)x + (product of roots) = 0
⇒ x² - { (a² - 8b) / 2a }x + (-2b) = 0
⇒ x² - { (a² - 8b) / 2a }x - 2b = 0
Multiply both sides by 2a
⇒ 2ax² - (a² - 8b)x - 4ab = 0
∴ The required quadratic equation is
2ax² - (a² - 8b)x - 4ab = 0
So, Option (D) - None, is correct.
Step-by-step explanation:
Hope this answer will help you.✌️