C1 is a circle of radius 2 touching x - axis and y - axis. C2 is another circle of radius greater than 2 and touching both the axes as well as the circle C1. Statement - 1 : Radius of circle C2 is \sqrt{2} \textrm{ } \left(\right. \sqrt{2} + 1 \left.\right) \textrm{ } \left(\right. \sqrt{2} + 2 \left.\right). Statement - 2 : Centres of both circles always lie on the line y = x.

Engineering

Mathematics

A Circle

Question

C₁ is a circle of radius 2 touching x - axis and y - axis. C₂ is another circle of radius greater than 2 and touching both the axes as well as the circle C₁.
Statement - 1 : Radius of circle C₂ is $\sqrt{2} (\sqrt{2} + 1) (\sqrt{2} + 2)$ .
Statement - 2 : Centres of both circles always lie on the line y = x.

A

Statement-1 is true, statement-2 is false.

B

Statement-1 is false, statement-2 is true.

C

Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

D

Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

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From the diagram,

$\sqrt{2} (r_{2} - r_{1}) = r_{2} + r_{1}$

but r₁ = 2

$\sqrt{2} (r_{2} - 2) = r_{2} + 2$

$(\sqrt{2} - 1) r_{2} = 2 + 2 \sqrt{2}$

$\Rightarrow r_{2} = \sqrt{2} (\sqrt{2} + 1) (\sqrt{2} + 2)$

Also centres of both the circles may also lie on y = – x.

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