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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