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Question

If two concentric circles are drawn with centre in first quadrant, one touches the X-axis and Y-axis and the other passes through the origin, the radius of the smaller circle is $$'r'$$, then the centre is


A
(2r,r)
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B
(r,2r)
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C
(r,r)
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D
none of these
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Solution

The correct option is C $$(r,r)$$

Let $$(a,b)$$ be the center of the circles

Given that one of the circles touches both the axes and the other passes through origin.

Smaller circle touches the axes and Bigger circle passes through $$O$$.

Given radius of smaller circle is $$r$$ and $$X,Y$$ axes are tangents

Which implies radius = perpendicular distance from center to tangent $$(y =0 , x = 0)$$

$$r = \dfrac{|a|}{\sqrt{1^2 + 0}} , r = \dfrac{|b|}{\sqrt{1^2 + 0}}$$

$$\implies a = r, b = r$$

$$\implies (a, b) = (r,r) = center$$


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