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A variable circle passes through the poi...

A variable circle passes through the point `A(a ,b)` and touches the x-axis. Show that the locus of the other end of the diameter through `A` is `(x-a)^2=4b y`.

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As shown in the figure, the variable circle passes through the point A(a,b) and touches the x-axis at ponit P.

Let the other end of diameter of the circle through A be B(h,k).
So, centre of the circle is `C((h+a)/(2),(k+b)/(2))`
Radius of the circle,` CP=(AB)/(2)`
`implies |(k+b)/(2)|=(sqrt((h-a)^(2)+(k-b)^(2)))/(2)`
Squaring and simplifying, we get `(h-a)^(2)=4bk`.
Therefore, required equation of the locus is `(x-a)^(2)=4by`.
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