To find the relationship between the chord lengths $x$ and $y$ and the angles they subtend at the center of a circle, we use the formula for the length of a chord:

Chord length = $2r \sin(\theta/2)$

where $r$ is the radius of the circle and $\theta$ is the angle subtended by the chord at the center.

Chord Calculations

For the first chord of length $x$, the angle subtended at the center is $60^{\circ}$.

$x = 2r \sin(60^{\circ}/2)$

$x = 2r \sin(30^{\circ})$

Since $\sin(30^{\circ}) = 1/2$:

$x = 2r \times (1/2)$

$x = r$

For the second chord of length $y$, the angle subtended at the center is $90^{\circ}$.

$y = 2r \sin(90^{\circ}/2)$

$y = 2r \sin(45^{\circ})$

Since $\sin(45^{\circ}) = 1/\sqrt{2}$:

$y = 2r \times (1/\sqrt{2})$

$y = \frac{2r}{\sqrt{2}}$

$y = \sqrt{2}r$

Relationship Between Chord Lengths

We found that $x = r$ and $y = \sqrt{2}r$.

Substitute $r$ with $x$ in the equation for $y$:

$y = \sqrt{2}x$

This matches Option A.