### Instruction: The graph of a parabola has the following properties:

$\bullet$ It passes through the point $(1,5).$

$\bullet$ The $y$-coordinate of the focus is 3.

$\bullet$ Its axis of symmetry is parallel to the $x$-axis.

$\bullet$ Its vertex lies on the $y$-axis.

Express the equation of the parabola in the form
\[ax^2 + bxy + cy^2 + dx + ey + f = 0,\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $c$ is a positive integer, and $\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$.### Response: Since the axis of symmetry is parallel to the $x$-axis, the parabola is a sideways parabola.
I recall that the equation of a sideways parabola with focus $(h,k)$ and directrix $x = l$ is \[(y - k)^2 = 4p(x - h),\]where $p$ is the distance from the focus to the vertex.
I need to find the values of $h,$ $k,$ $l,$ and $p.$
Since the vertex lies on the $y$-axis, $h = 0.$
Since the $y$-coordinate of the focus is 3, $k = 3.$
The parabola passes through $(1,5),$ so plugging in those values, I get \[(5 - 3)^2 = 4p(1 - 0).\]
Simplifying, I get $4 = 4p,$ so $p = 1.$
The distance from the focus to the directrix is also $p,$ so the directrix is $x = -1.$
Now I have all the information I need to write the equation of the parabola.
It is \[(y - 3)^2 = 4(x - 0),\]or equivalently, \[y^2 - 6y + 9 = 4x.\]
To put it in the required form, I move everything to one side and get \[-4x + y^2 - 6y + 9 = 0.\]
I check that the coefficients satisfy the given conditions.
They are integers, $c = 1$ is positive, and the greatest common divisor of the absolute values of the coefficients is 1.