Question

the difference between squares of two numbers of 120. the squares of smaller number is twice to greater number. find the numbers.

Answer 1

The answer is given below :

Let us consider that the numbers are p and q with p > q.

Then, by the given conditions,

Difference of the squares of p and q = 120

=> p² - q² = 120 .....(i)

and

Square of the smaller number = Twice the greater number

=> q² = 2p .....(ii)

Putting q² = 2p in (i), we get

p² - 2p = 120

=> p² - 2p + 1 = 120 + 1

=> (p - 1)² = 121

=> (p - 1)² = 11²

=> p - 1 = ± 11

=> p = 1 ± 11

Thus, we get

p = 1 + 11 and, p = 1 - 11

i.e., p = 12 and p = -10

When, p = 12, from (ii), we get

q² = 2 × 12

=> q² = 24

=> q² = 2² × 6

=> q = ± 2√6

Again, when p = -10, from (ii), we get

q² = 2 × (-10)

=> q² = -20

=> q² = 2² × 5 × i², where i = √(-1) and i = -1

=> q = ± 2√5 i

Hence, the required numbers are

(12, ± 2√6) and (-10, ± 2√5 i).