Question
A satellite dish has a parabolic cross-section. The dish is deep at the centre and wide at the rim. Where should the receiver be placed to catch all incoming parallel signals? (Practical problem in design)
Solution — Step by Step
Place the vertex at the origin with the axis along the -axis. The standard form is , where is the focal length — exactly where the receiver goes.
The rim is at height (depth) with (half-width). Plug in:
The focus lies at , i.e., above the vertex along the axis.
Place the receiver above the centre of the dish. All parallel signals (e.g., from a distant satellite) will reflect off the parabolic surface and converge there.
Final answer: The receiver goes at the focus, above the dish vertex.
Why This Works
The defining property of a parabola is that any ray parallel to its axis reflects off the curve and passes through the focus. This is why headlights, telescopes, and satellite dishes all use parabolic shapes.
The equation encodes this geometry — once you know two points on the parabola, is determined, and so is the focus.
Alternative Method
Use the focus-directrix definition: any point on the parabola is equidistant from the focus and the directrix. Here, with , the directrix is the line , and any signal-reflection geometry stays consistent.
JEE problems often ask for the latus rectum length: . Here that’s — equal to the dish width at the focus level. Common PYQ pattern.
Common Mistake
Confusing “depth” with “height of focus”. The depth is the -coordinate of the rim, not the focus position. Always set up the equation first, solve for , and then locate the focus.