Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then

Engineering

Mathematics

Common Tangent To two Circle

Question

$\sqrt{a}, \sqrt{b}, \sqrt{c} are in A . P .$

a, b, c are in A.P.

$\frac{1}{\sqrt{a}} = \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{c}}$

$\frac{1}{\sqrt{b}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{c}}$

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Length of direct common tangent for

circle C₁ and C₂ is

$AB = \sqrt{{(a + b)}^{2} - {(a - b)}^{2}}$

For C₂ and C₃

Length of direct common tangent is

$BC = \sqrt{{(a + c)}^{2} - {(a - c)}^{2}}$

For C₁ and C₃

Length of direct common tangent is

$AC = \sqrt{{(b + c)}^{2} - {(b - c)}^{2}}$

AB + BC = AC

$\sqrt{{(a + b)}^{2} - {(a - b)}^{2}} + \sqrt{{(a + c)}^{2} - {(a - c)}^{2}} = \sqrt{{(b + c)}^{2} - {(b - c)}^{2}}$

$\sqrt{ab} + \sqrt{ac} = \sqrt{bc}$

$\frac{1}{\sqrt{c}} + \frac{1}{\sqrt{b}} = \frac{1}{\sqrt{a}}$