A very bright young student of mine who has not learned trigonometry presented me with this question she came up with.
A unit circle is placed at the origin. At any point along the circle we can draw a tangent line and calculate its slope. Then we can plot (x-coordinate, slope). It seems to make this nice curved X shape. Can we make an equation for it?
Your task is to find an equation, without trigonometry.
Note: As interesting as it was to try drawing the shape on a whiteboard, I had to fight the urge to use trig for this equation. In fact, I think it has a nice parametric form using trig, so you could also find that. Or whatever you like. We were just playing around with some math.
[Edit: I misinterpreted x-coordinate as x-intercept, so disregard most of this post]
Will plot the x-coordinate on the horizontal axis; the slope on the vertical axis
x^2 + y^2 = 1
(a,b) is a point on the circle.
Tangent line is: y = -(a/b)x + b + a^2/b
x-intercept is : a + b^2/a
Slope = -a/b
and a^2 + b^2 = 1
Locus of points: (x,y) = (a + b^2/a , -a/b)
You can put this in terms of one parameter, for example a, but this loses some of values by forcing x or y to be negative or positive; unless you put a ± in front of each term.
Slope in terms of a:
-a/√(1-a^2)
x-intercept in terms of a:
a + (1-a^2)/a = 1/a^2
Locus of points: (x,y) = (± 1/a^2 , ± a/√(1-a^2))
Edited on February 19, 2024, 4:19 pm
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Posted by Larry
on 2024-02-19 12:06:12 |
Doing some algebra
