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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


Solution

The equation of variable circle can be 
$$(x-0)^2 (y - b)^2 = b^2$$     $$[\because x - a$$ is a tangent to circle]
$$\Rightarrow (2018 - a)^2 + (2009 - b)^2 = b^2$$   ....(1)
$$4-a = a - 2018$$ and $$ k - b = b - 2009$$      [as $$(4, k)$$ and $$(2018, 2008)$$ are diametrically opposite points]
$$\Rightarrow 4 = 2a - 2018$$ and $$k = 2b - 2009$$
$$\Rightarrow a = \dfrac{h + 2018}{2} $$     ...(2)
$$\Rightarrow b = \dfrac{k + 2009}{2}$$     ...(3)
using (2) and (3) in (1) we get
$$\left(\dfrac{2018-h}{2}\right)^2 + \left(\dfrac{2009 - k}{2}\right)^2 = \left(\dfrac{k + 2009}{2}\right)^2$$
$$\Rightarrow (20018 - h)^2= (k+2009)^2 - (k - 2009)^2$$
$$\Rightarrow (4 - 20018)^2 = (2k)(2\times 2009)$$
$$\Rightarrow (h - 2018)^2 = 4\times 2009 \times k$$
comparing with eqn, $$(x - x_0)^2 = 4ay$$ we get, that
$$x_0 = 2018$$ and $$a = 2009$$
the locus required is a parabola.

1368813_1177614_ans_b112267a05cd42c8b06dd8c672b44de6.png

Mathematics

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