Science:Math Exam Resources/Courses/MATH110/April 2019/Question 04 (b)
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Question 04 (b) 

Suppose the equation given above [] describes a curve in the plane. Find the equation of the tangent line to such curve at the point . 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

The tangent line at a point in the curve has equation
In this problem and the value of is to be determined using the equation of the curve. After this just substitute these values in the equation for computed in part (a). 
Checking a solution serves two purposes: helping you if, after having used the hint, 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. We will first calculate the value of as in the hint. Since the point is a point on the curve, it must satisfy the equation of the curve. Therefore (since we have ), must satisfy
Expanding this the equation becomes
This has a solution when either or . The former case yields , while the latter is not possible because the left hand side is always strictly positive. Therefore, the unique point in the curve having is the point . The slope of the line is obtained by evaluating the formula for obtained in part (a) at this point:
By substituting these values in the tangent line equation given in the hint, we obtain the equation
Rearranging this, we conclude that the equation of the tangent line to the curve at is . 