Question

A variable chord is drawn through the origin to the circle $x^2+y^2-2ax=0$. The locus of the centre of the circle drawn on this chord as diameter is

• A. $x^2+y^2+ax=0$
• B. $x^2+y^2+ay=0$
• C. $x^2+y^2-ax=0$
• D. $x^2+y^2-ay=0$

Answer 1

The correct option is C $x^2+y^2-ax=0$

Given

$\Rightarrow x^2+y^{{}^2}-2ax=0$

$\Rightarrow x^2+y^{{}^2}+a^2-2ax-a^2=0$

$\Rightarrow {(x-a)}^2+y^2=a^2$

So the center circle is $O(a,0)$ and radius is $=a$

Let the center of chord drown through be $p(h,k)$

Let other end of chord be $(c,b),$

So,$(\frac{c+0}{2},\frac{b+0}{2})=(h,k)$

$\Rightarrow c=2h,b=2k$

But,point $(c,b)$ lies on the circle

$\Rightarrow x^2+y^2-2ax=0$

So,${\left(2h\right)}^2+{\left(2k\right)}^2-2a.2h=0$

$\Rightarrow h^2+k^2-ah=0$

Hence It is locus $x^2+y^2-ax=o$