Will trigonometry help?

Algebra Level 3

Positive real numbers $x, y$ satisfy the equations $x^2+y^2 = 1$ and $x^4+y^4 = \dfrac{17}{18}$ . If $xy = \dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

The answer is 7.

1 solution

Akeel Howell
Mar 25, 2017

By squaring $x^2+y^2$ we see that $2x^2y^2 = (x^2+y^2)^2-(x^4+y^4) = \dfrac{1}{18}$ .

$\therefore xy = \dfrac{1}{6} \implies a+b = 1+6 = 7$ .

Why will trigo help?? Oh x^2 + y^2 =1 thats why?

Md Zuhair - 4 years, 2 months ago

I don't think that would be very helpful in solving the problem. Even if we used $\sin{\theta}$ and $\cos{\theta}$ instead of $x$ and $y$ the solution would likely follow the same idea of squaring and subtracting.

Akeel Howell - 4 years, 2 months ago

I nwver opposed your solution... I did the same way. But I just asjed why you named the problem will trigo help

Md Zuhair - 4 years, 2 months ago

@Md Zuhair – Yeah, I wasn't finding a name for it and the sum of squares made me think of the Pythagorean identity.

Akeel Howell - 4 years, 2 months ago

Will trigonometry help?

The answer is 7.

1 solution

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