Question

Use the Distance Formula to write an equation of the parabola. focus: \((0,-2)\) directrix: \(y=2\)

Short answer

The equation of the parabola is \(x^2+8y = 0\).

Step-by-step solution

Applying the distance formula I

Here, we are given that the focus of the parabola is at \( (0,-2) \) and the directrix is the line \( y = 2 \). We mark a point, say \( (x,y) \) on the parabola. Then we have to set up the equation for the distance from \( (x,y) \) to the focus \( (0,-2) \) using the distance formula, which is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }\). This gives us the first equation \( \sqrt{(x-0)^2 + (y-(-2))^2} = \sqrt{x^2+(y+2)^2} \)

Applying the distance formula II

The distance from \( (x,y) \) to the directrix \( y = 2 \) is simply the vertical distance between \( y \) and 2, which is \( |y-2| \), and is given by the absolute value of the difference in the y-coordinates. But since the parabola is below the directrix here, we don’t need absolute value and can simply take the difference \( (2-y) \). This yields the second equation \( 2-y \).

Equating the two distances

Since the distance from the focus to a point on the parabola is the same as the distance from the point to the directrix, we solve the equation \( \sqrt{x^2+(y+2)^2} = 2-y \)

Simplifying the equation

Square both sides of the equation to get rid of the square root, \( x^2+(y+2)^2 = (2-y)^2 \). This simplification leads to \( x^2+y^2+4y+4 = 4-4y+y^2\). We can cancel \(y^2\) on both sides and bring all terms to one side to find \(x^2+8y+4-4 = 0\). That further simplifies to \(x^2+8y = 0\), which is the equation of the parabola.

Key concepts

Distance Formula

Understanding the distance formula is crucial when working with the geometry of shapes, particularly conic sections such as parabolas. The distance formula \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \) is derived from the Pythagorean Theorem and computes the straight-line distance between two points in a coordinate plane.

For example, if we have a focus at point A with coordinates \( (x_1, y_1) \) and a random point B on the parabola with coordinates \( (x_2, y_2) \) the distance from A to B is given by this formula. The point is, no matter where B lies on the parabola, its distance to the focus remains constant, which helps form the characteristic 'U' shape of the parabola.

This becomes a backbone concept when we set up an equation for a parabola, as it relates every point on the curve directly to its focus and directrix.

Focus of a Parabola

The focus of a parabola is a fixed point from which the distance to any point on the parabola is equal to the distance from that point to a line called the directrix. It's the defining feature of a parabola and helps determine its width and orientation.

For a parabola described by the equation \(y=ax^2\text{+bx+c}\), the focus lies along the axis of symmetry of the parabola. This axis is a vertical line for parabolas that open up or down, and a horizontal line for those that open left or right. The exact position of the focus influences the 'sharpness' of the parabola—closer to the vertex means a sharper curve, while farther away means a flatter one.

Knowing the focus allows us to use the distance formula to generate the parabolic shape by ensuring all points equidistant from the focus and directrix lie on the curve.

Directrix of a Parabola

Alongside the focus, a parabola has another defining feature known as the directrix. The directrix is a fixed line and it doesn't touch the parabola itself. It runs parallel to the axis of symmetry and lies opposite to the opening direction of the parabola.

The directrix serves as a reference line from which every point on the parabola is equidistant to the focus. While the focus is a point, the directrix is a horizontal or vertical line, typically expressed in the simple form \(y=k\) or \(x=k\).

The distance from any point on the parabola to the directrix is measured perpendicularly to the directrix, which simplifies into a linear expression when substituted into the distance formula. This relationship is vital for deriving the equation of a parabola.

Conic Sections Algebra

Conic sections are the curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas, each with a unique set of algebraic properties.

A parabola can be expressed as an algebraic equation generally written in the form \(y=ax^2+bx+c\) for a vertical orientation or \(x=ay^2+by+c\) for a horizontal orientation. The coefficients of these terms determine the parabola's shape and position in the coordinate plane.

Algebraic manipulation of conic sections involves leveraging formulas and operations rooted in Euclidean geometry and coordinate algebra. In particular, for parabolas, the distance from the focus to any point on the curve equals the perpendicular distance from that same point to the directrix. We use this principle to derive the equation of a parabola, solving for constants that describe its specific shape and location.