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Question

A variable circle passes through the point A(2018,2009) and touches the x axis,then the locus of the other end of the diameter through A is a/an

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Solution

The equation of variable circle can be
(x0)2(yb)2=b2 [xa is a tangent to circle]
(2018a)2+(2009b)2=b2 ....(1)
4a=a2018 and kb=b2009 [as (4,k) and (2018,2008) are diametrically opposite points]
4=2a2018 and k=2b2009
a=h+20182 ...(2)
b=k+20092 ...(3)
using (2) and (3) in (1) we get
(2018h2)2+(2009k2)2=(k+20092)2
(20018h)2=(k+2009)2(k2009)2
(420018)2=(2k)(2×2009)
(h2018)2=4×2009×k
comparing with eqn, (xx0)2=4ay we get, that
x0=2018 and a=2009
the locus required is a parabola.

1368813_1177614_ans_b112267a05cd42c8b06dd8c672b44de6.png

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