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

MATH110 April 2013

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[hide] 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.

[show] Hint 1
Lets suppose that the curves intersect each other. What does it mean to intersect at right angles?

[show] 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.

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

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[show] Solution

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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