Problem 99 - Algebra and Trigonometry

Question

Write the standard form of the equation of a parabola whose points are equidistant from \(y=4\) and \((-1,0)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \((x + 1)^2 = -8(y - 2)\)

Step 1: Determine the focus and directrix

From the problem we know that the focus of the parabola is at \((-1, 0)\) and the directrix is the line \(y = 4\).

Step 2: Formulate the general equation

From the standard equation of a parabola, \(4p(y - k) = (x - h)^2\) where \((h, k)\) is the vertex of the parabola, and \(p\) is distance from the vertex to the focus or to the directrix. If directrix is \(y = 4\) and the focus is at \((-1, 0)\), the vertex is the midpoint between the directrix and the focus, which is at the point \((-1, 2)\). Hence \(h = -1\) and \(k = 2\). The value of \(p\), the distance from the vertex to the focus or directrix is equal to 2. Note that the parabola opens downwards since its focus is above the directrix.

Step 3: Write the equation of the parabola

Substitute values of \(h\), \(k\), and \(p\) into the equation. We get: \[4(2)(y - 2) = (x + 1)^2\], simplifying we get: \[(x + 1)^2 = -8(y - 2)\].