The equation of the circle which is touched by y $$=$$ x, has its centre on the positive direction of the $$x-axis$$ and cuts off a chord of length $$2$$ units along the line $$3y$$−$$x=0$$, is

Question

The equation of the circle which is touched by y $$=$$ x, has its centre on the positive direction of the $$x-axis$$ and cuts off a chord of length $$2$$ units along the line $$3y$$−$$x=0$$, is

Infinity Learn · Accepted Answer

Since the required circle has its centre on $$x-axis$$, So, let the coordinates of the centre be (a$$, $$0). The circle touches y $$=$$ x. Therefore,Radius $$=$$ Length of the . from (a$$, $$0) on x $$-y$$ $$=$$ $$0$$⇒  Radius $$=a2The$$ circle cuts off a chord of length $$2$$ units along x−$$3y=0$$∴ $$a22=12+a$$−$$3$$×$$012+(3)22$$⇒$$a22=1+a24$$⇒$$a=2Thus$$, centre of the circle is at (2$$, $$0) and radius $$=a2=2Hence$$, its equation is $$x2+y2$$−$$4x+2=0$$

The equation of the circle which is touched by $y = x$ , has its centre on the positive direction of the $x - a x i s$ and cuts off a chord of length 2 units along the line $\sqrt{3} y - x = 0,$ is

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The equation of the circle which is touched by y = x, has its centre on the positive direction of the x-axis and cuts off a chord of length 2 units along the line 3y−x=0, is

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The equation of the circle which is touched by $y = x$ , has its centre on the positive direction of the $x - a x i s$ and cuts off a chord of length 2 units along the line $\sqrt{3} y - x = 0,$ is