MathDB

[asy]draw(Circle((0,0), 1)); dot((0,0)); label("

O

", (0,0), S); label("

A

", (-1,0), W); label("

B

", (1,0), E); label("

a

", (-0.5,0), S); draw((-1,-1.25)--(-1,1.25)); draw((1,-1.25)--(1,1.25)); draw((-1,0)--(1,0)); draw((-1,0)--(-1,0)+2.3*dir(30)); label("

C

", (-1,0)+2.3*dir(30), E); label("

D

", (-1,0)+1.8*dir(30), N); dot((-1,0)+.4*dir(30)); label("

E

", (-1,0)+.4*dir(30), N); [/asy] In this figure

AB

is a diameter of a circle, centered at

O

, with radius

a

. A chord

AD

is drawn and extended to meet the tangent to the circle at

B

in point

C

. Point

E

is taken on

AC

so that

AE=DC

. Denoting the distances of

E

from the tangent through

A

and from the diameter

AB

x

and

y

, respectively, we can deduce the relation:

\text{(A)}\ y^2=\dfrac{x^3}{2a-x} \qquad \text{(B)}\ y^2=\frac{x^3}{2a+x}\qquad \text{(C)}\ y^4=\frac{x^2}{2-x}\qquad\\ \text{(D)}\ x^2=\dfrac{y^2}{2a-x}\qquad \text{(E)}\ x^2=\frac{y^2}{2a+x}

Problem Statement