Q 51. - Calculus

Question

Find an equation for the curve obtained as the set of midpoints of every chord with one endpoint at (1, 0) on the unit circle r = 1. Express your answer 

(a) with parametric equations.

(b) in rectangular coordinates.

(c) in polar coordinates. 

Step-by-Step Solution

Answer

Part (a) The parametric equations are $x=\frac{1}{2}+\frac{1}{2}\cos t,y=\frac{1}{2}\sin t$

Part (b) The rectangular coordinates are ${\left(x-\frac{1}{2}\right)}^2+4y^2=1$

Part (c) $r=\cos \theta$

Part (a) Step 1: Given information

Part (a) Step 2: Concept

A collection of quantities is defined as a function of one or more independent variables called parameters in a parametric equation.

Part (a) Step 3: Calculation

Consider a unit circle $r=1$ with one endpoint at $(1,0)$

Every chord's midpoints are given, along with one end endpoint $(1,0)$

Trigonometric functions are included in the parametric equations. With a radius of one,

Thus the equations are,

$x=\frac{1}{2}+\frac{1}{2}\cos t\text{ and }y=\frac{1}{2}\sin t$

Therefore, the parametric equations are $x=\frac{1}{2}+\frac{1}{2}\cos t,y=\frac{1}{2}\sin t$

Part (b) Step 1: Calculation

Consider the parametric equations $x=\frac{1}{2}+\frac{1}{2}\cos t$ and $y=\frac{1}{2}\sin t$

Take $x=\frac{1}{2}+\frac{1}{2}\cos t$

Subtract $-\frac{1}{2}$ from sides of the equation.

$$\begin{gathered} x-\frac{1}{2}=-\frac{1}{2}+\frac{1}{2}\cos t \\ x-\frac{1}{2}=\cos t \dots \left(1\right) \end{gathered}$$

Then

$x-\frac{1}{2}=\cos t$

Now take $y=\frac{1}{2}\sin t$

Multiply by 2 on both sides of the equation.

$2y=\sin t\dots \left(2\right)$

To eliminate the parameter $t$, square the equations and add them.

${\left(x-\frac{1}{2}\right)}^2={\cos }^{2}t,4y^2={\sin }^{2}t$

Thus,

$$\begin{gathered} {\left(x-\frac{1}{2}\right)}^2+4y^2={\cos }^{2}t+{\sin }^{2}t \\ {\left(x-\frac{1}{2}\right)}^2+4y^2={\cos }^{2}t+{\sin }^{2}t \\ {\left(x-\frac{1}{2}\right)}^2+4y^2=1 \end{gathered}$$

Therefore, the rectangular coordinates are ${\left(x-\frac{1}{2}\right)}^2+4y^2=1$

Part (c) Step 1: Explanation

In polar coordinates the equation is given by $r=\cos \theta$ Therefore the answer is $r=\cos \theta$