construction, locus, starting with perpendicular diameteres and tangents

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construction, locus, starting with perpendicular diameteres and tangents

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#1 h Aug 23, 2018, 1:49 PM • 2 Y

Y by Adventure10, Mango247

$AB$ and $CD$ are perpendicular diameters of a circle. $L$ is the tangent to the circle at $A$ . $M$ is a variable point on the minor arc $AC$ . The ray $BM, DM$ meet the line $L$ at $P$ and $Q$ respectively. Show that $AP\cdot AQ = AB\cdot PQ$ .
Show how to construct the point $M$ which gives $BQ$ parallel to $DP$ .
If the lines $OP$ and $BQ$ meet at $N$ find the locus of $N$ .
The lines $BP$ and $BQ$ meet the tangent at $D$ at $P'$ and $Q'$ respectively. Find the relation between $P'$ and $Q$ '.
The lines $D$ P and $DQ$ meet the line $BC$ at $P"$ and $Q"$ respectively. Find the relation between $P"$ and $Q"$ .

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#2 h Jun 14, 2019, 8:40 PM • 1 Y

Y by Adventure10

Circle with equation $x^{2}+y^{2}=r^{2}$ .
Points $A(r,0),B(-r,0),D(0,-r)$ , the line $l\ :\ x=r$ .

Point $M(r\cos \alpha,r\sin \alpha)$ .

Point $P(r,2r\tan \frac{\alpha}{2})$ .
Point $Q(r,\frac{2r\sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2}})$ .

The line $OP\ :\ y=2\tan \frac{\alpha}{2} \cdot x$ cuts the line $BQ\ :\ y=\frac{\sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2}} \cdot (x+r)$
in the point $N\ :\ x_{N}=\frac{r}{1-2\tan \frac{\alpha}{2}}$ .

Eliminating the parameter $\alpha$ , we find the locus $y=x-r$ with $x \geq r$ .

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