The third proportional to 7 and 63 is:

The correct option is 567.

To find the third proportional to two numbers, we use the concept of continued proportion. If three numbers, let's say $a$, $b$, and $x$, are in continued proportion, then the ratio of the first to the second is equal to the ratio of the second to the third.

This relationship is written as:

$a:b=b:x$

In fraction form, this is expressed as:

$\frac{a}{b}=\frac{b}{x}$

By cross-multiplying, we can solve for the third proportional, $x$:

$a\times x=b^2$

$x=\frac{b^2}{a}$

Here, the given numbers are $a=7$ and $b=63$. Substituting these values into the formula, we get:

$x=\frac{{63}^2}{7}$

We can simplify this by writing ${63}^2$ as $63\times 63$:

$x=\frac{63\times 63}{7}$

Since $63$ divided by $7$ is $9$, the equation simplifies to:

$x=9\times 63$

Calculating the product:

$x=567$

Therefore, the third proportional to 7 and 63 is 567.