Problem: 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. If the coordinates of E are (x,y), then:
Answer Choices:
A. y2=2a−xx3
B. y2=2a+xx3
C. y4=2a−xx2
D. x2=2a−xy2
E. x2=2a+xy2
Solution:
2ax=ACAE=ACCD and BC2=CD⋅CA and yx=BC2a so that BC2=x24a2y2. Since 2ax⋅BC2= ACCD⋅CD⋅CA,2ax⋅x24a2y2=CD2∴x2ay2=CD2=AE2=x2+y2∴y2(2a−x)=x3∴y2=2a−xx3
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