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Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Translate a polar curve into a Cartesian equation.
- Justify the conclusion "(x−1)²+y²=1" using the stated assumptions.
Before you start
Polar coordinates and completing the square.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Identify the curve r=2cos θ.
Why this math matters
Translate a polar curve into a Cartesian equation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The equation describes the standard polar locus.
- The origin is included even though its angle is not unique.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Recognize a circle hidden in a polar equation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Identify the curve r=2cos θ.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Multiply by r: r²=2r cos θ
This creates expressions with familiar Cartesian counterparts.
Work through the mathematics
x²+y²=2x
Substitute r²=x²+y² and r cos θ=x.
Check the conclusion
(x−1)²+y²=1
The locus is a radius-one circle centered at (1,0), including the origin.
The result
(x−1)²+y²=1
The locus is a radius-one circle centered at (1,0), including the origin.
Common mistakes to catch
- A simple polar formula need not be centered at the polar origin.
- Polar coordinates of a point are not unique.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Where does the curve meet the positive x-axis farthest from the origin?
Show a hint
Set θ=0.
Reveal answer and explanation
(2,0)
r=2 there.
Practice 2
Is the center the polar origin?
Show a hint
Read the completed-square equation.
Reveal answer and explanation
No
The center is shifted one unit along x.
Take the idea with you
Choose a coordinate system that simplifies a shifted circular boundary.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Eliminate a parameter without losing traversal information
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