#### Answer

$x^2+y^2=65$

#### Work Step by Step

RECALL:
The standard form of a circle whose center is at $(h, k)$ and radius $r$ is $(x--h)^2+(y-k)^2=r^2$.
The center is at $(0, 0)$, then $h=0$ and $k=0$.
Thus, the standard form of the given circle's equation of the circle is:
$(x-0)^2+(y-0)^2=r^2
\\x^2+y^2=r^2$
The circle passes through the point $(4, 7)$.
This means that the coordinates of this point satisfy the equation of the circle.
Substitute the x and y values of the point into the equation to obtain:
$x^2+y^2=r^2
\\4^2+7^2=r^2
\\16+49=r^2
\\65=r^2$
Thus, the standard form of the given circle's equation is:
$x^2+y^2=65$