How to solve the "four" variables problem

Question

Given x, y, z, and w are real numbers which satisfy these three equation: $$x^{2} + 5z^{2} = 10$$ $$yz - xw = 5$$ $$xy + 5zw = \sqrt{105}$$

Find the value of $y^{2} + 5w^{2}$

Can anyone give a hint? I don't even know where to start

Answer 1

Hint:

$$5(yz-xw)^2+(xy+5zw)^2=5(5^2)+105=230$$

Now $$5y^2z^2+5x^2w^2+x^2y^2+25z^2w^2=(5z^2+x^2)(y^2+5w^2)$$

Answer 2

Method$\#1:$

Solve the two simultaneous equations in $x,z$

$$z=\dfrac{aw+5y}{y^2+5w^2},x=\dfrac{ay-25w}{y^2+5w^2}$$

Replace the values of $z,w$ in $$10=x^2+5z^2$$

$$10=\dfrac{a^2y^2+625w^2-50ayw+5(a^2w^2+25y^2+10ayw)}{(y^2+5w^2)^2}$$

$$=\dfrac{(a^2+125)y^2+(625+5a^2)w^2}{(y^2+5w^2)^2}=\dfrac{a^2+125}{y^2+5w^2}$$

where $a^2=105$

See also : Brahmagupta-Fibonacci Identity

Method$\#2:$

WLOG $x=\sqrt{10}\cos t, z=\sqrt2\sin t$

$$yz - xw = 5\implies y(\sqrt2\sin t) - (\sqrt{10}\cos t)w = 5$$ $$xy + 5zw = \sqrt{105}\implies 5(\sqrt2\sin t)w+ (\sqrt{10}\cos t)y = \sqrt{105} $$

Solve for $\cos t,\sin t$ and use $\cos^2t+\sin^2t=1$ to eliminate $t$