A designer is drawing the cross-section of a headlight reflector…

A designer is drawing the cross-section of a headlight reflector. She places the axis of the reflector along the $x$-axis, with the lamp end at the origin, and measures $x$ and $y$ in centimetres. The curve must have a constant sub-normal. The sub-normal is the length along the $x$-axis from the foot of the perpendicular, dropped from a point of the curve, to the point where the normal meets the axis. It is $3\ \text{cm}$ at every point. The curve lies above the axis, so $y > 0$, and it passes through the point $(6, 6)$. She writes $y\dfrac{dy}{dx} = 3$ and integrates it to get $y^2 = 6x + C$. Find the constant $C$, the height $y$ of the curve at $x = 24\ \text{cm}$, and the distance of the focus of the parabola from the vertex, in centimetres. The parabola $y^2 = 4ax$ has its focus at distance $a$ from the vertex.