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8.If A and G are the arithmetic and geometric means respectively of two numbers thenthat the numbers are A+VA2 -G2 and A VA2 G2proveNCERT) |
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Answer» AM of number a and b = (a + b)/2 GM of number a and b = √ab Here, AM : GM = m : n (a + b)/2 : √ab = m : n (a + b)/2√ab = m/n Applying componendo and dividendo rule,(a + b + 2√ab)/(a + b - 2√ab) = (m + n)/(m - n)(√a² + √b² + 2√ab)/(√a² + √b² -2√ab) = (m + n)/(m - n) (√a + √b)²/(√a - √b)² = (m + n)/(m - n) take square root both sides,( √a + √b )/(√a - √b) = √(m + n)/√(m - n) again applying componendo and dividendo,( √a + √b + √a - √b)/(√a + √b - √a + √b) = {√(m + n) + √(m - n)}/{√(m + n) - √(m - n) }2√a/2√b = {√(m + n) + √(m - n)}/{√(m + n) - √(m - n) }√a/√b = {√(m + n) + √(m - n)}/{√(m + n) - √(m - n) }taking square both sides, a/b =[{√(m + n) + √(m - n)}/{√(m + n) - √(m - n) }]²a/b = {m + n + m - n - 2√(m² - n²)}/{m + n + m - n -2√(m² - n²)}a/b = {2m + 2√(m² - n²)}/{2m - 2√(m² - n²)}a/b = {m + √(m² - n²)}/{m - √(m² - n²)} hence, a : b = m + √(m² - n²) : m - √(m² - n²) |
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