91Ó°ÊÓ

The Distance Formula is a fundamental concept used to calculate the distance between two points in a plane. It is derived from the Pythagorean theorem and is written as \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.
• The difference \((x_2 - x_1)\) gives the horizontal distance, and \((y_2 - y_1)\) gives the vertical distance between the points.

In the context of the parabola exercise, the Distance Formula is used to find the distance between the point \( P(x, y) \) and the focus \( (-2, 3) \). This involves substituting the coordinates into the distance formula, resulting in \[ \sqrt{(x + 2)^2 + (y - 3)^2} \]. This formula is essential for showing the relationship of points forming a parabola.

Conic Sections are curves obtained by intersecting a double-napped cone (imagine two cones with their points touching) with a plane. The four basic types of conic sections are ellipses, circles, parabolas, and hyperbolas. These shapes are foundational in geometry and are widely applicable in various scientific fields.

• Ellipse: An elongated circle with two focal points.
• Circle: A special type of ellipse where the two foci are at the same point.
• Parabola: Formed when a plane cuts parallel to the edge of a cone.
• Hyperbola: Created when a plane intersects both napes of the cone.

In our exercise, we're dealing with a parabola, a unique type of conic section characterized by the property that every point on it is equidistant from a point known as the focus and a line called the directrix. Recognizing how these properties influence the parabolic shape helps in understanding its equation.

The Equation of a Parabola defines the set of all points equidistant from a point called the focus and a line known as the directrix. In this exercise, we have a parabola with focus at \((-2, 3)\) and directrix \(x = 2\). To derive the equation, set up the equality between the distances: \[ |x - 2| = \sqrt{(x + 2)^2 + (y - 3)^2} \]. Squaring both sides removes the square root and absolute value, simplifying it to: \[ (x - 2)^2 = (x + 2)^2 + (y - 3)^2 \].Solving this provides \[ y = -(x + 2)^2 + 6 \]. This equation signifies a parabola that opens downward, with its vertex shifted horizontally and vertically based on the transformations applied to the standard parabola equation \( y = ax^2 \). Understanding how to set up and simplify these equations is crucial for graphing and analyzing parabolas in both algebraic and geometric contexts.