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  • If PN is the perpendicular from a point on a rectangular hyperbola to its asymptotes, then the locus of the mid−point of PN isOption: 1 circle<

If PN is the perpendicular from a point on a rectangular hyperbola to its asymptotes, then the locus of the mid−point of PN is

Option: 1

circle


Option: 2

parabola 

 


Option: 3

ellipse


Option: 4

hyperbola 

 


Answers (1)

best_answer

Let\mathrm{x y=c^2} be the rectangular hyperbola, and let \mathrm{P\left(x_1, y_1\right)} be a point on it. . Let Q (h, k) be the mid-point of PN. 

\mathrm{\text { Then the coordinates of } Q \text { are }\left(x_1, \frac{y_1}{2}\right)}

\mathrm{\therefore \quad x_1=h \text { and } \frac{y_1}{2}=k \Rightarrow x_1=h \text { and } y_1=2 k}

\mathrm{\text { But }\left(x_1, y_1\right) \text { lies on } x y=c^2}

\mathrm{\therefore \quad h \cdot(2 k)=c^2 \Rightarrow h k=c^2 / 2 \text {. }}

 

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shivangi.bhatnagar

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