If a conic section is written as a polar equation, and the denominator involves sin theta, what conclusion can be drawn about the directrix?

illusiia 2020-11-11 Answered
If a conic section is written as a polar equation, and the denominator involves sin θ, what conclusion can be drawn about the directrix?
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hosentak
Answered 2020-11-12 Author has 100 answers
IF a conic has a focus at the origin, directrix y=± p and with a positive real number e as the eccentricity, then the polar equation of the conic is given by r=ep1 ± e sinθ So when there is sinθ in the denominator, the directrix is the form y= ± p i.e. it is parallel to x-axis.
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Why is this solution incorrect?
Prove that on the axis of any parabola there is a certain point K which has the property that, if a chord PQ of the parabola be drawn through it, then 1PK2+1QK2 is the same for all positions of the chord.
If it would be valid for standard parabola than it would be valid for all parabolas. Thus, proving for y2=4ax.
let the point K be (c,0)
Equation of line PQ using parametric coordinates: x=c+rcosθ (Equation 1), y=rsinθ (Equation 2)
From equation 1 and 2: (xc)2+y2=r2 (Equation 3)
Using Equation 3 and y2=4ax, we get this quadratic in r: r24ax(xk)2=0 (Equation 4)
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From Equation 4, r1+r2=0 and r1r2=(4ax+(xk)2)
We know, 1PK2+1QK2=1r12+1r22=(r1+r2)22r1r2(r1r2)2=2r1r2
As value of r1r2 is not constant thus 1PK2+1QK2 does not turn out to be constant. Hence, this solution is incorrect.
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