locus of points C such that in ABC a side is k times it's corresponding altitude

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locus of points C such that in ABC a side is k times it's corresponding altitude

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Source: 1st UNESCO Competition 1995-6 p6

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#1 h Jan 17, 2021, 11:19 PM • 1 Y

Y by Mango247

Given a segment $AB$ and a constant $k>0$ . Find the locus of points $C$ in the plane such that in triangle $ABC$ , the length of a side is $k$ times the length of the corresponding altitude.

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#2 h Jan 23, 2021, 8:45 PM • 1 Y

Y by teomihai

Given $A(0,0),B(b,0)$ and $C(x,y)$ then $b=k \cdot y$ , the easy case.

Given $A(0,0),B(b,0),C(p,q)$ , point $D(\frac{bp^{2}}{p^{2}+q^{2}},\frac{bpq}{p^{2}+q^{2}}$ and $D \in AC$ and $BD \bot AC$ .

$BD=\frac{bq}{\sqrt{p^{2}+q^{2}}}$ and $AC=\sqrt{p^{2}+q^{2}}$ .
Locus is the circle $x^{2}+y^{2}-kby=0$ .

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