Let two numbers have arithmetic mean 9 and geometric mean 4. Then thes...
Let the two numbers be α and β. Given that
∴ Required equation is x
2 – 18x + 16 = 0
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Let two numbers have arithmetic mean 9 and geometric mean 4. Then thes...
Solution:
Given, arithmetic mean = 9 and geometric mean = 4.
Let the two numbers be a and b.
Arithmetic mean = (a + b)/2 = 9
Geometric mean = √ab = 4
Squaring both sides of the second equation, we get:
ab = 16
Multiplying both sides of the first equation by 2, we get:
a + b = 18
Squaring both sides of this equation, we get:
a2 + b2 + 2ab = 324
Substituting ab = 16, we get:
a2 + b2 + 32 = 324
a2 + b2 = 292
Multiplying both sides of the equation a + b = 18 by (a – b), we get:
a2 – b2 = 324 – 4ab
Substituting ab = 16, we get:
a2 – b2 = 260
Adding the equations a2 + b2 = 292 and a2 – b2 = 260, we get:
2a2 = 552
a2 = 276
a = √276 = 2√69
Substituting this value of a in the equation a + b = 18, we get:
2√69 + b = 18
b = 18 – 2√69
Therefore, the two numbers are:
a = 2√69 and b = 18 – 2√69
The quadratic equation with these roots is:
(x – 2√69)(x – (18 – 2√69)) = 0
Expanding this equation, we get:
x2 – 20x + 16 = 0
Hence, the correct answer is option D.
Let two numbers have arithmetic mean 9 and geometric mean 4. Then thes...
Let numbers be x,y now average mean=(x+y)/2 =9 therefore x+y=18 and y=18-x. --------------------(equation 1) geometric mean=√xy=4 xy=16 substituting value of y from equation 1 we get x(18-x)=16 18x-x^2=16 x^2-18x+16 therefore the solution of this equation will give us the numbers hence option d is correct