Problem 56 - Algebra 2

Question

Writing Explain how to find the distance from the focus to the directrix of the parabola \(x=2 y^{2} .\)

Step-by-Step Solution

Answer

The distance from the focus to the directrix of the parabola \(x=2 y^{2}\) is 1 unit.

Step 1: Finding the coefficient 'p' from the given equation of the parabola

To find p, equate the coefficient of \(y^2\) in the given equation to 4p, or 2 = 4p.

Step 2: Calculate the value of 'p'

To find the value of 'p', divide the coefficient of \(y^2\) by 4, that is \(p = 2/4 = 0.5\)

Step 3: Calculate the distance from the focus to the directrix

Distance between the focus to the directrix of the parabola is measured as 2*p, which gives 2*0.5 = 1.

Key Concepts

Distance formula

One of the essential tools in geometry is the distance formula. It helps calculate the distance between two points on a Cartesian coordinate plane. The formula is derived from the Pythagorean theorem and is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \(x_1, y_1)\) and \(x_2, y_2)\) are the coordinates of the two points. This formula is incredibly useful in various areas of mathematics, including analyzing parabolas in conic sections. When applied to parabolas, especially those oriented along the axes, the formula doesn't serve to find a simple point-to-point distance but rather helps in understanding geometric properties like the length between the focus and the directrix.

Focus and directrix

The concept of focus and directrix is fundamental in understanding the geometry of a parabola. A parabola is defined as the set of all points equidistant from a fixed point, called the focus, and a line, called the directrix. Key points:

• The focus is a point through which every line perpendicular to the directrix and passing through the lateral surface of the parabola reflects back to.
• The directrix is a line that helps maintain the shape and position of the parabola in relation to the focus.

In mathematical terms, the distance from the focus to any point on the parabola is equal to the perpendicular distance from that point to the directrix. This balance of distances fundamentally defines a parabola and makes the parameters of the focus-directrix a significant aspect in conic sections. When given a parabola like \(x = 2y^2\), calculating the distance from the focus to the directrix involves first identifying the parameter \(p\), using the relationship in the equation format, which leads us to the solution.

Conic sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These include circles, ellipses, parabolas, and hyperbolas. Parabolas are unique among conics because they possess a distinctive focus-directrix property that governs their shape. In terms of a parabola's equation, which typically appears in the form \(x = 4py^2\) or \(y = 4px^2\), the parameter \(p\) is vital. It represents the distance from the vertex to the focus, and, equivalently, to the directrix.

• The vertex is the point where the parabola changes direction, acting as the midpoint between the focus and the directrix.
• Varying \(p\) changes the width and orientation of the parabola.

Understanding these core characteristics of conics provides insight into not only geometrical properties but also practical applications, such as satellite dishes and reflecting telescopes, whose parabolic shapes are chosen due to their focal properties.