Problem 341 - Calculus Volume 3

Question

Find the Cartesian equation describing the given shapes. A parabola with focus (2,-5) and directrix \(x=6\)

Step-by-Step Solution

Answer

The parabola is described by the equation \(y^2 + 10y + 8x - 7 = 0\).

Step 1: Understand parabola definition

A parabola is defined as the set of all points equidistant from a point (focus) and a line (directrix). In this problem, the focus is located at \((2, -5)\) and the directrix is the vertical line \(x = 6\).

Step 2: Determine the vertex

The vertex of the parabola is midway between the focus and the directrix. Calculate the x-coordinate of the vertex as the midpoint between x of the focus (2) and the directrix line \(x = 6\). Thus, vertex \((h, k)\) is at \((\frac{2+6}{2}, -5) = (4, -5)\).

Step 3: Calculate distance

The distance from the focus to the directrix is \(6 - 2 = 4\). This means the parabola opens horizontally. The distance from the vertex to the focus (\(p\)) is half of this, so \(p = 2\). Since the vertex is to the left of the directrix, the parabola opens leftwards, meaning \(p\) is \(-2\).

Step 4: Write the standard form

The standard form for a horizontally oriented parabola is \((y - k)^2 = 4p(x - h)\). Substitute \(h = 4\), \(k = -5\), and \(p = -2\). This gives the equation: \((y + 5)^2 = -8(x - 4)\).

Step 5: Simplify to Cartesian form

Expand and simplify the equation \((y + 5)^2 = -8(x - 4)\) to the form \(ax^2 + bxy + cy^2 + dx + ey + f = 0\). Expanding gives \(y^2 + 10y + 25 = -8x + 32\), or rearranged, \(y^2 + 10y + 8x - 7 = 0\).

Key Concepts

Understanding a Parabola

A parabola is a fascinating mathematical shape that appears in various fields, including physics and engineering. It is defined as the set of all points that are equidistant from a fixed point, known as the focus, and a fixed line, called the directrix.

The shape of the parabola depends on the position of its focus and directrix. For example, if the focus is vertically above or below the directrix, the parabola will open upwards or downwards. In essence, the parabola's alignment is influenced by its focus and directrix. Recognizing these components helps in graphing and understanding the parabola's properties.

Some real-world examples of parabolas include the paths followed by projectiles in motion under gravity and the design of satellite dishes and reflective telescope mirrors.

Focus and Directrix Relationship

In a parabola, the role of the focus and directrix is crucial, as they set the foundation of the shape.

• **Focus**: The focus is a specific point inside the parabola from which every point on the parabola is equidistant to the directrix.
• **Directrix**: This is a straight line outside the parabola. Any point on the parabola is equidistant from this line and the focus.

For a parabola with focus \(2, -5\) and directrix \(x = 6\), the parabola is horizontally aligned. In this case, the focus-directrix relationship determines the orientation and the specific shape of the parabola.

The equation of the parabola can be derived using these features, helping students visualize how each point on the parabola maintains its equidistance from both the focus and directrix.

Vertex of Parabola

The vertex of a parabola is a pivotal point that determines the parabola's symmetry and direction.

Here, the vertex is halfway between the focus and the directrix. For the example with focus \(2, -5\) and directrix \(x = 6\), the vertex is calculated as the midpoint of the x-coordinates of the focus and the directrix. This gives the vertex at \( (4, -5) \).

• **Coordinate Calculation**: Use the formula for finding the midpoint on the x-axis, so \( h = rac{2+6}{2} = 4 \).
• **Y-Coordinate**: Since the parabola opens horizontally, the y-coordinate of the vertex stays the same as the focus, \(-5\).

Understanding the vertex helps in graphing the parabola and proves essential in transforming the parabola into different mathematical representations, like the vertex form.

Horizontally Oriented Parabola

A horizontally oriented parabola differs from the more commonly encountered vertical parabolas. These parabolas open to the right or left.

In the given problem, the parabola opens to the left because the focus is to the left of the directrix. This orientation affects its equation and characteristics:

• **Equation Form**: The equation of a horizontally oriented parabola in standard form is \( (y - k)^2 = 4p(x - h) \).
• **Directionality**: Here, \( p \) determines whether the parabola opens rightward (positive value) or leftward (negative value). In this case, \( p = -2 \), thus opening left.

This equation captures the relationship between the parabola's geometric properties and its algebraic representation, which is particularly useful in solving real-world problems involving parabolic paths or trajectories.