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

Answer 1

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 $k=2b-2009$

$\Rightarrow a=\frac{h+2018}{2}$ ...(2)

$\Rightarrow b=\frac{k+2009}{2}$ ...(3)

using (2) and (3) in (1) we get

${\left(\frac{2018-h}{2}\right)}^2+{\left(\frac{2009-k}{2}\right)}^2={\left(\frac{k+2009}{2}\right)}^2$

$\Rightarrow (20018-h{)}^2=(k+2009{)}^2-(k-2009{)}^2$

$\Rightarrow (4-20018{)}^2=(2k\left)\right(2\times 2009)$

$\Rightarrow (h-2018{)}^2=4\times 2009\times k$

comparing with eqn, $(x-x_0{)}^2=4ay$ we get, that

$x_0=2018$ and $a=2009$

the locus required is a parabola.