A parabola can be represented by the equation $y^2 = 12x$.
Which equation represents the directrix?
The equation given is \(y^2 = 12x\). This equation is in the standard form of a horizontal parabola with its vertex at the origin \((0,0)\), which is written as \(y^2 = 4px\). The parameter \(p\) determines the geometric properties of the parabola, such as the position of the focus and the directrix. In this equation, the coefficient of \(x\) is 12, so we set \(4p = 12\). Solving for \(p\), we divide both sides by 4 to get \(p = 3\).
For a parabola defined by the standard equation \(y^2 = 4px\), the focus is located at the point \((p, 0)\) and the directrix is a vertical line defined by the equation \(x = -p\). Since we have calculated \(p = 3\), we substitute this value into the formula for the directrix. Thus, the equation of the directrix is \(x = -3\).
Engineers and designers often use parabolic shapes in the construction of satellite dishes or car headlights because of their reflective properties. To design these components correctly, they need to know the focus and directrix of the parabolic cross-section. For a parabolic mirror described by the equation \(y^2 = 12x\), identifying that the directrix lies at \(x = -3\) helps them calculate the optimal placement for the receiver or light source to ensure efficient energy collection or projection.