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Northstar
Lv 7
Northstar asked in Science & MathematicsMathematics · 1 decade ago

Polar Coordinates – Equation of Locus of Points?

A line segment of length 2a slides with its ends on the x and y-axes. Find the polar equation of the locus described by the point P(r, θ) at which the perpendicular from the origin intersects the moving line.

Update:

That's r = a sin(2θ).

Update 2:

I can see by carefully plotting points that the answer apperas to be asin(2θ), but I am interested in how to derive the answer.

1 Answer

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  • nle
    Lv 7
    1 decade ago
    Favorite Answer

    r* ( tan(theta) + cot(theta) ) =2a

    or

    r= 2 a/ ( tan(theta) + cot(theta) )

    EDIT : proof is easy

    call the line AB

    AP =OP tan(theta)

    BP = OP cot(theta)

    add

    AP + BP= OP (tan(theta) +cot(theta) )

    2a = r ( tan(theta) +cot(theta) )

    which is reduced to your equation

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