A circle C 1 of radius r 0 rolls externally around C 2 : x 2 + y 2 + 2 r x = 0 ; their line of centres has inclination 6 0 ∘ to the positive x -axis. Find an equation of a direct common tangent.

Question

A circle C 1 ​ of radius r > 0 rolls externally around C 2 ​ : x 2 + y 2 + 2 r x = 0 ; their line of centres has inclination 6 0 ∘ to the positive x -axis. Find an equation of a direct common tangent.

TestHub · Accepted Answer

The centres are ( − r , 0 ) and ( 0 , 3 ​ r ) , with radii r . A direct common tangent is parallel to their line of centres, so its normal is ( 3 ​ , − 1 ) . At ( − r , 0 ) , distance r requires ∣ k − 3 ​ r ∣ = 2 r , hence k = r ( 3 ​ ± 2 ) , option B. The plus signs in A and B coincide, but their minus-sign lines differ.

Mathematics - Circle Question with Solution | TestHub

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