Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 09 (c)
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Question 09 (c) 

The curve is known as a Limaçon of Pascal.
(c) Explain why, for every point on the Limaçon with and tangent line of slope , there must exist another point on the Limaçon with and tangent line of slope . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Look for symmetry in the expression of the slope. 
Hint 2 

What happens when we replace by ? 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. By part (b), we have where we write in place of to emphasize that it is the slope at the point .
Note that if a point is on the Limaçon, then so is . We can easily show this by plugging into the equation of the curve. By Hint 2, we observe that This means that at the point the slope has the opposite sign of the slope at . In other words, if , then at the other point , the slope is . The point is not the same point as because . Also, at the point , the coordinate is . Answer: . 