Science:Math Exam Resources/Courses/MATH110/April 2013/Question 04

MATH110 April 2013

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Question 04
Let a and b be constants. Show that if the curves xy = a and x2 - y2 = b intersect each other, they do so at right angles.

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
Lets suppose that the curves intersect each other. What does it mean to intersect at right angles?

Hint 2
Two curves intersect at right angles if their tangent lines at the point of intersection have slopes whichs are negative reciprocals of each other. Thus, this suggests taking derivatives and checking that at this intersection point, the slopes are negative reciprocals.

Hint 3
To find the slope of the tangent lines use implicit differentiation.

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Using implicit differentiation, we solve for the derivative of y in both curves ${\begin{aligned}xy&=a\\y+xy'&=0\\y'&={\frac {-y}{x}}\end{aligned}}$ and for the second curve, ${\begin{aligned}x^{2}-y^{2}&=b\\2x-2yy'&=0\\2(x-yy')&=0\\x-yy'&=0\\y'&={\frac {x}{y}}\end{aligned}}$ These two slopes are negative reciprocals of each other. Hence, if the curves intersect each other, they do at a right angle.

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