Expert: Paul Klarreich - 7/24/2009

Question
Let Ps=(Xs,Ys) be the point on the unit circle s counterclockwise units away from the point (1,0). Let r(x)=ax^2+bx+c be a quadratic function. Also let Qs be the point r(s) units away from the positive vertical line passing. (1,0). Let f(s) be the x coordinate of the x intercept of the line passing through Qs and Ps. Determine the limit of f(s) as s approaches 0 from the right.  

Ok, let me answer some of your questions. S is the arc length of the circle. And the vertical line is passing through the point (1,0). And Qs is a point on the vertical line passing through (1,0). So the coordinates for Qs is (1,ax^2+bx+c)

Answer
Questioner: Sharon
Country: United States
Category: Advanced Math
Private: No
Subject: Math 2
Question: Let Ps=(Xs,Ys) be the point on the unit circle s counterclockwise units away from the point (1,0). Let r(x)=ax^2+bx+c be a quadratic function. Also let Qs be the point r(s) units away from the positive vertical line passing. (1,0). Let f(s) be the x coordinate of the x intercept of the line passing through Qs and Ps. Determine the limit of f(s) as s approaches 0 from the right.  

Ok, let me answer some of your questions. S is the arc length of the circle. And the vertical line is passing through the point (1,0). And Qs is a point on the vertical line passing through (1,0). So the coordinates for Qs is (1,ax^2+bx+c)


P(xs,ys) is at an arc distance from (1,0).  Conclusion:

xs = cos s,   ys = sin s  [you work it out.]

Qs is on the line x = 1, and at a distance r(s) = as^2 + bs + c.

So Qs has coordinates  (1,as^2 + bs + c).

The line has slope:

m = (r(s) - sin s)/(1 - cos s)

The equation of the line is:

y - sin s =  (r(s) - sin s)/(1 - cos s)*(x - cos s)

Now choose  y = 0.

- sin s =  (r(s) - sin s)/(1 - cos s)*(x - cos s)

- sin s(1 - cos s) =  (r(s) - sin s)(x - cos s)

cos s sin s - sin s = (r(s) - sin s)x - (r(s) - sin s)cos s

cos s sin s - sin s = (r(s) - sin s)x - r(s) cos s + sin s cos s

- sin s = (r(s) - sin s)x - r(s) cos s

r(s) cos s - sin s = (r(s) - sin s)x
   r(s) cos s - sin s
x = ------------------
    (r(s) - sin s)


As  s --> 0,  r(s) --> c, as in  as^2 + bs + c

So this approaches:

   c cos 0 - sin 0
x = ---------------- = c/c = 1
    c - sin 0