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# MEI FP3 envelopes watch

1. can someone please explain how to form the equation of a family of curves in the form f(x,y,p)=0.

For example, in the MEI FP3 book the first example is: stick a pin into a piece of paper, place one edge of a ruler against the pin and draw a short, straight line, using the other edge of the ruler. Do this repeatedly turning the ruler slightly around the pin each time.

Treat the centre of the paper, where the pin is as the origin of the x,y plane. The ruler has a width of 2.5 cm, and the angle made between the x-axis and the line perpendicular to the ruler line is phi. find the equation of the line in terms of x, y and phi.

Any help would be great because this is driving me crazy!!

By the way, the solution should be xcos(phi) + ysin(phi) = 2.5
2. (cos(phi), sin(phi)) is normal to the line. So if (x0, y0) is any point on the line then

(x - x0, y - y0).(cos(phi), sin(phi)) = 0

for all (x, y) on the line. So the line has equation

x cos(phi) + y sin(phi) = k

for some constant k. Draw a picture to find k.
3. I can see that Jonny solved this, but I was wondering why I'm getting a slightly different answer.

(See attachment.)

The equation of the line is y=mx+c.

c = 2.5/cos(90-phi) = 2.5/sin(phi)

m = tan(90-phi) = cot(phi)

So:
y = cot(phi)x + 2.5/sin(phi)
=> y sin(phi) - x cos(phi) = 2.5

Am I misinterpreting the question?
Attached Images

4. Ah, reading over it again, I think I've taken the wrong angle as phi - my phi should be 180-phi. This gives the right answer.
5. cheers

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Updated: April 11, 2006
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