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#1 h Apr 1, 2016, 12:30 PM • 2 Y

Y by Adventure10, Mango247

Let $AB$ be a chord on parabola $y=x^2$ and $AB||Ox$ . For each point C on parabola different from $A$ and $B$ we are taking point $C_1$ lying on the circumcircle of $\triangle ABC$ such that $CC_1||Oy$ . Find a locus of points $C_1$ .

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#2 h Apr 4, 2016, 10:06 PM • 2 Y

Y by Adventure10, Mango247

Let $A, B, C,C_1$ have coordinates $(a,a^2),(-a,a^2),(c,c^2),(c,d)$ and $O(x_0,y_0)$ is center of circumcircle $\triangle ABC$
$(x-x_0)^2+(y-y_0)^2=r^2$ -equation of circle.
For $A,B$ we get $(a-x_0)^2+(a^2-y_0)^2=(-a-x_0)^2+(a^2-y_0)^2$
$(a-x_0)^2=(-a-x_0)^2$ => $x_0=0$
For $A,C$ we get $a^2+(a^2-y_0)^2=c^2+(c^2-y_0)^2$
$(a^2-c^2)(1+a^2+c^2-2y_0)=0$ => $2y_0=1+a^2+c^2$
For $C,C_1$ we get $c^2+(c^2-y_0)^2=c^2+(d-y_0)^2$
$c^2+d-2y_0=0$
$d=2y_0-c^2=1+a^2$

So answer is line $y=1+a^2$ where $a$ - x-coordinate of $A$

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