Science:Math Exam Resources/Courses/MATH110/December 2010/Question 05 (c)
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Question 05 (c) 

Two curves are said to intersect orthogonally at x =a if (i) they intersect at x = a and (ii) their tangent lines are perpendicular at x = a. Consider the curve . Find constants b and c such that the curve intersects it orthogonally at . 
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 

To solve this question, you will need to meet the two conditions given in the definition of orthogonal intersection. It might be easiest to start with the second condition  that the tangent slopes of each curve be perpendicular to each other. What is the tangent slope of the first function you're given? What does the tangent slope of the second function have to be, if it's going to be the negative reciprocal of that first slope? 
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. To find a solution for , we need to satisfy two conditions: (1) and must intersect at and (2) the tangent line of at must be perpendicular to the tangent line of at . For this problem it doesn't matter which condition you test first; in this case, we will start with the second condition and then proceed to the first condition. We begin by finding the slope of the tangent line to each curve at . For the derivative is and so the slope at is . For the derivative is and so the slope at is . To satisfy condition (2), these two slopes must be perpendicular, i.e. negative reciprocals of each other. Since the tangent slope of is 1, this means we need the tangent slope of to equal to 1. However, we know the tangent slope of is , so this means . Having found we now turn to the first condition of the definition to find . The first condition requires that and intersect at . This means that must equal . In equations, this is:
So . Thus the curve satisfies both of our conditions and intersects orthogonally the curve at . 