Problem 2 - Algebra 2

Question

Write an equation for a graph that is the set of all points in the plane that are equidistant from the given point and the given line. $$ F(0,-1), y=1 $$

Step-by-Step Solution

Answer

The equation of the parabola is \(y = x^2/4\)

Step 1: Understanding Parabola The definition

A parabola is the locus of all points in a plane that are equidistant from a fixed point, the focus F, and a fixed line, the directrix. Here it is provided that the focus \(F(0, -1)\) and the directrix is \(y = 1\).

Step 2: Formulate the Equation

Let's take a point \(P(x, y)\) on the parabola. According to the definition, the distance from \(P\) to \(F\) is equal to the distance from \(P\) to the line \(y=1\). So, we have the equation: \(\sqrt{(x-0)^2 + (y-(-1))^2} = |y-1|\). Square both sides to eliminate the square root.

Step 3: Simplify the Equation

Squaring both sides we get: \(x^2 + (y+1)^2 = (y-1)^2\). We simplify to obtain the equation of the parabola as : \(y = x^2/4\).

Key Concepts

Distance Formula

The Distance Formula helps us calculate how far apart two points are in a plane. In simpler words, it measures the length of the shortest path connecting them. The formula is derived from the Pythagorean Theorem and is written as:

• For two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \(d\) between them is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In this problem, we apply the Distance Formula to ensure points on the parabola are exactly as far from the focus point as they are from the directrix, a condition that defines a parabola.
By squaring both sides of the equation, the square root is removed, simplifying the process of finding a parabola's equation. This squaring step aligns with algebraic methods, making calculations manageable and straightforward.

Focus and Directrix

The focus and directrix are essential elements of a parabola. The focus is a fixed point inside the curve, while the directrix is a fixed line outside of it. They help us locate the set of all points that form the curve of a parabola.
Here is how they work together:

• The focus, in this instance \( F(0, -1) \), is the point that every point on the parabola is equidistant from the directrix, here the line represented by \( y = 1 \).
• The directrix itself acts as a reference line. Every point on the parabola maintains an equal distance from both the focus and this line.

Understanding the relationship between the focus and directrix is crucial. They determine the orientation and the shape of the parabola. Where the focus is placed actively affects how 'wide' or 'narrow' the parabola appears, or even if it opens upwards or downwards.

Equation of a Parabola

Writing the equation of a parabola is about expressing the geometric definition into a usable algebraic form. Based on its definition, the equation stems from the equality of distances: the focus to the point \( P(x, y) \) and the directrix to the same point on the curve.

• To form this, we start with the equation: \[ \sqrt{(x-0)^2 + (y-(-1))^2} = |y-1| \]

Through algebra, we square both sides to remove the square root and get: \[ x^2 + (y+1)^2 = (y-1)^2 \]. Simplifying this yields a cleaner form: \[ y = \frac{x^2}{4} \].
This equation tells us much about the parabola's properties, such as its vertex and orientation. The absence of a linear \( x \) term in the final form suggests the parabola is symmetric about the y-axis, and since it opens upwards, we can verify that our parabola corresponds correctly to the problem's conditions.