A parabolic reflector is 20cm in diameter and 5cm deep. Which one of the following is co ordinates of its focus? a) (0, 5) b) (0, 5) c) (5, 0) d) ( 5, 0)

The parabolic reflector is a parabola opening downwards with its vertex at the top. Given:

• Diameter = 20 cm ⇒ Radius $a=10$ cm
• Depth = 5 cm ⇒ Vertex at $(0,5)$, focus at $(0,k)$ where $k<5$.

The standard equation of a downward-opening parabola with vertex at $(0,b)$ is: $$x^2=-4p(y-b)$$ Here, the parabola passes through the point $(10,0)$ (edge of the reflector). Substituting: $$10^2=-4p(0-5)⟹100=20p⟹p=5$$ The focus lies at a distance $p$ below the vertex, so its coordinates are: $$(0,5-p)=(0,0)$$ However, the given options suggest the parabola is oriented differently. Reconsidering the standard form for a parabola with vertex at $(0,5)$ and focus at $(0,5-p)$, and using the depth (5 cm) as the distance from vertex to focus: $$p=5⟹\text{Focus}=(0,5-5)=(0,0)$$ But this does not match any options. Revisiting the problem, the reflector is 5 cm deep, meaning the focus is 5 cm below the vertex along the y-axis. Thus, the correct focus coordinate is: $$\boxed{(0,0)}$$ However, the options imply a different interpretation. If the parabola is 5 cm deep from the rim to the vertex, then the vertex is at $(0,5)$, and the focus is at $(0,5-p)$. Using the standard relation for a parabola $x^2=4py$, the focus is at $(0,p)$. Here, the depth (5 cm) corresponds to $p=1.25$ cm (since the rim is at $(10,0)$, and substituting gives $100=4p(5)⟹p=5$). Thus, the focus is at: $$\boxed{(0,0)}$$ But this still does not match the options. The correct interpretation is that the vertex is at the origin (0,0) and the parabola opens downward with depth 5 cm. The standard equation is: $$x^2=-4py$$ Given the rim is at $(10,-5)$, substituting: $$10^2=-4p(-5)⟹100=20p⟹p=5$$ Thus, the focus is at $(0,-p)=(0,-5)$.

Final Answer: $\boxed{(0,-5)}$ (Option b).