Calc question

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Discus: SAT/ACT Tests and Test Preparation: March 2004 Archive: Calc question
 By Aim78 (Aim78) on Wednesday, March 10, 2004 - 10:32 pm: Edit

This is a homework question (not worth a grade) that's pretty tough. Help me out here:

A circle of radius 2 and center (0,0) can be parametrized by the equations x=2cost and y=2sint. Show that for any value of t, the line tangent to the circle at (2cost, 2sint) is perpendicular to the radius.

It seems common sense that a tangent line would be perpendicular to the radius, but how do I show it algebraically?

 By Vsage3 (Vsage3) on Wednesday, March 10, 2004 - 10:46 pm: Edit

if the radius at any time t has a slope that can be expressed by 2sint / 2cost, then dy/dx can be expressed by 2cost / -2 sint, or -1/(2sint / 2cost). How is that hard :p

 By Aim78 (Aim78) on Wednesday, March 10, 2004 - 11:20 pm: Edit

Ah-ha...so I get how you find the slope of the radius, but what did you take the derivative of?

 By Vsage3 (Vsage3) on Wednesday, March 10, 2004 - 11:24 pm: Edit

here's the logic behind it. If x = 2cost and y = 2sint, then y / x = 2sint / 2cost. To find dy/dx, we need to take the derivative of y and x separately and condense it, i.e. dy/dx = (dy/dt)/(dx/dt). You do know how to get dy/dt and dx/dt, so you just take the derivative of what y = and what x = and you get -x/y.

 By Aim78 (Aim78) on Thursday, March 11, 2004 - 12:45 am: Edit

I got it. When you take dy/dt/dx/dt, the dt's cancel and you're left with dy/dx. Thanks!