Question

12. The sum of two irrational numbers multiplied by the larger one is 70 and their difference
is multiplied by the smaller one is 12; the two numbers are
a) 372, 273
b) 572,315 c) 272,572 d) none of these.​

Answer 1

Answer:

d) none of these.​

Step-by-step explanation:

Let the two irrational numbers be √a and √b, where a and b are positive integers but not a perfect square with √a > √b

By the given conditions,

√a (√a + √b) = 70

or, a + √(ab) = 70 ..... (1)

√b (√a - √b) = 12

or, √(ab) - b = 12 .... (2)

Now {(1) - (2)} gives

a + b = 58

or, a = 58 - b

Putting a = 58 - b in (1), we get

58 - b + √{b (58 - b)} = 70

or, 58 - b + √(58b - b²) = 70

or, √(58b - b²) = b + 12

or, 58b - b² = b² + 24b + 144

or, 2b² - 34b + 144 = 0

or, b² - 17b + 72 = 0

or, b² - 9b - 8b + 72 = 0

or, b (b - 9) - 8 (b - 9) = 0

or, (b - 9) (b - 8) = 0

Either b - 9 = 0 or, b - 8 = 0

This gives b = 9, 8

Since b is not a perfect square, we take b = 8

Then a = 58 - 8 = 50

So √a = √50 = 5√2

and √b = √8 = 2√2

Therefore the two irrational numbers are 5√2 and 2√2

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