Question

Parabola with vertex \((2,3)\) and focus \((2,5)\)

Short answer

The equation of the parabola is \((x - 2)^2 = 8(y - 3)\).

Step-by-step solution

Identify the Orientation

The vertex of the parabola is at the point \((2, 3)\) and the focus is at \((2, 5)\). Since both the x-coordinates of the vertex and the focus are the same, the parabola opens vertically. Additionally, the focus is above the vertex, so the parabola opens upwards.

Calculate the Distance to Obtain the Value of 'p'

The distance between the vertex and the focus \(p\) is calculated using the y-coordinates because the parabola opens vertically. \[p = 5 - 3 = 2\]

Write the Vertex Form of the Equation

Since the parabola opens upwards and has a vertex at \((2, 3)\), the equation of the parabola in vertex form is:\[ (x - h)^2 = 4p(y - k) \]Substitute \(p = 2\), \(h = 2\), and \(k = 3\):\[ (x - 2)^2 = 8(y - 3) \]

Convert to Standard Form (Optional)

To express the equation in the standard form for better readability or if required:First, expand the equation, \((x-2)^2 = 8(y-3)\):\[ x^2 - 4x + 4 = 8y - 24 \]Rearrange and simplify:\[ x^2 - 4x - 8y + 28 = 0 \]This is the standard form of the parabola equation.

Key concepts

Parabola

A parabola is a U-shaped curve that can open either upwards or downwards (if oriented vertically) or sideways (if oriented horizontally). It is defined as the set of all points that are equidistant from a fixed point called the focus and a line called the directrix.

• Key properties include the vertex, which is the highest or lowest point of the parabola depending on its orientation.
• The axis of symmetry is a line that vertically or horizontally divides the parabola into two mirror images.
• A parabola can open in four directions: up, down, left, or right, determined by its orientation and the position of the vertex relative to the focus.

Understanding the different forms of parabolic equations will help in identifying these characteristics and the specific orientation of any given parabola.

Vertex Form of Equation

The vertex form of a parabola's equation is particularly useful because it provides direct information about the parabola's vertex. The general vertex form is:\[ (x-h)^2 = 4p(y-k) \]

• Here, \(h,k\) represents the vertex of the parabola.
• The term \(p\) indicates the distance from the vertex to the focus (along the axis of symmetry).

This form makes graphing easier because the vertex \(h,k\) is easily identified, which is especially helpful when sketching the parabola. For a parabola opening vertically, the expression \(x - h)^2\) ensures the curve is symmetrical around the vertical line \(x=h\).

Focus of a Parabola

The focus of a parabola is a crucial component in its geometric definition. It is the point from which the set of distances equating to the parabola's curve is measured. The distance from any point on the parabola to the focus is equal to the perpendicular distance from the point to the directrix.

• For a vertically oriented parabola, the focus lies either above or below the vertex depending on whether the parabola opens upwards or downwards.
• In this context, a parabola with a focus at \(2,5\) and vertex at \(2,3\) opens upwards because the focus is located above the vertex.

This structural information about the focus helps in determining the direction in which the parabola opens and also the equation of the parabola.

Standard Form of Equation

The standard form of a parabola's equation is another way to express its shape and orientation. The equation \ \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) is often used to illustrate the parabola's general orientation and position.

• For parabolas, the B and C coefficients typically simplify such that either the \(x^2\) or \(y^2\) term is present, but not both, indicating the lack of rotation relative to the axes.
• In the context of the given problem, the parabola's standard form \(x^2 - 4x - 8y + 28 = 0\) offers an expanded expression that can be useful in solving further algebraic manipulations or integrations in analytic geometry.

Understanding how to convert between these forms provides flexibility in solving different types of problems involving parabolas.