Let S1 and S2 be the two circles. S1 is tangent to x - axis and S2 is tangent to y - axis and the straight line y = mx (m > 0) touches both the circles at their common point. If centre of the circle S1 is (3, 1) then radius of the circle S2 is \frac{p}{q}, p, q ∈ N. Find the least value of (p + q).

Engineering

Mathematics

A Circle

Chord of Contact Chord with given Mid Point and Pair of Tangents

Common Tangent To two Circle

Question

Let S₁ and S₂ be the two circles. S₁ is tangent to x - axis and S₂ is tangent to y - axis and the straight line y = mx (m > 0) touches both the circles at their common point. If centre of the circle S₁ is (3, 1) then radius of the circle S₂ is $\frac{p}{q}$ , p, q ∈ N. Find the least value of (p + q).

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C₁C₂ = r + 1

$\sqrt{(r - 3)^{2} + 4} = r + 1$

$(r - 3)^{2} + 4 = (r + 1)^{2} = r^{2} + 2 r + 1$

$\Rightarrow 8 r = 12 \Rightarrow r = \frac{3}{2} \equiv \frac{p}{q}$

∴ p + q = 5.