Problem 17 - The Calculus with Analytic Geometry

Question

Find a polar equation of the graph having the given cartesian equation.\(y^{2}=4(x+1)\)

Step-by-Step Solution

Answer

The polar equation is: \[ r = \frac{4 \cos \theta + 4}{\sin^2 \theta} \]

Step 1: Convert Cartesian Equation to Standard Form

The given Cartesian equation is: \[ y^{2} = 4(x + 1) \] Reorganize it to recognize it as a parabola standard form: \[ y^{2} = 4a(x - h) \] Here, it compares to \[ y^{2} = 4(x + 1) \] indicating \( a = 1 \) and the vertex at \( (h, k) = (-1, 0) \).

Step 2: Express in Terms of Polar Coordinates

Using polar coordinates \( x = r\cos\theta \) and \( y = r\sin\theta \), substitute them into the equation: \[ (r\sin\theta)^{2} = 4(r\cos\theta + 1) \]

Step 3: Simplify Polar Equation

Distribute and simplify the equation: \[ r^2 \sin^2 \theta = 4r \cos \theta + 4 \] Divide both sides by \( r \): \[ r \sin^2 \theta = 4 \cos \theta + \frac{4}{r} \] Further simplifying, the expression becomes: \[ r = \frac{4 \cos \theta + 4}{\sin^2 \theta} \]

Key Concepts

Cartesian to polar conversion

Converting coordinates from Cartesian to polar involves using specific relationships between the two systems. Cartesian coordinates \(x, y\) are transformed to polar coordinates \(r, \theta\) using these equations: \( x = r \cos\theta \) and \( y = r \sin\theta \). Here, \( r \) represents the distance from the origin to the point, and \( \theta \) denotes the angle from the positive x-axis to the point. For the given equation \( y^{2} = 4(x + 1) \), we substitute the polar coordinates for \( x \) and \( y \) leading to the equation \( (r \sin\theta)^{2} = 4(r \cos\theta + 1) \). This substitution helps in converting the provided Cartesian equation into its corresponding polar form, which is easier for some types of analyses or plotting.

parabola in polar coordinates

A parabola in polar coordinates follows a different form compared to Cartesian. The equation \( y^{2} = 4(x + 1) \) represents a parabola that opens towards the right. To better understand this, let's consider the standard form for a horizontal parabola: \( y^{2} = 4a(x - h) \). Identifying parameters: comparing \( y^{2} = 4(x + 1) \) implies \( a = 1 \) and the vertex at \( (h, k) = (-1, 0) \). Transforming this equation into polar coordinates and expanding yields \( (r \sin\theta)^{2} = 4(r \cos\theta + 1) \). By simplifying, we effectively describe the parabola in a different yet equivalent representation that can be easily used in various applications.

simplifying polar equations

After substituting the polar coordinates into the Cartesian equation, the next step is simplifying the polar equation. Starting with: \( r^2 \sin^2 \theta = 4r \cos \theta + 4 \), we can divide both sides by \( r \) (assuming \ r \eq 0 \). This gives \( r \sin^2 \theta = 4 \cos \theta + \frac{4}{r} \). Further simplification leads to: \( r = \frac{4 \cos \theta + 4}{\sin^2 \theta} \, which consolidates the equation into a simpler form. Such simplification steps are crucial as they make the equation more manageable and can reveal underlying properties about the geometric figure, making further calculus or algebraic operations more straightforward.