If distances from origin to centres of three circles {x^2} + {y^2}

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10-1.Circle and System of Circles

hard

If the distances from the origin to the centres of the three circles ${x^2} + {y^2} - 2{\lambda _i}\,x = {c^2},(i = 1,\,2,\,3)$ are in $G. P.$, then the lengths of the tangents drawn to them from any point on the circle ${x^2} + {y^2} = {c^2}$ are in

$A. P.$

$G. P.$

$H. P.$

None of these

Solution

(b) Obviously $\lambda _2^2 = {\lambda _1}{\lambda _3}$.

Now let any point $( – c,\;0)$be on ${x^2} + {y^2} = {c^2}$.

So length of tangents on the circles ${x^2} + {y^2} – 2{\lambda _i}x – {c^2} = 0$ from $( – c,\;0)$ are $2{\lambda _1}c,\;$ $2{\lambda _2}c$ and $2{\lambda _3}c$.

Hence these are also in $G.P.$

Standard 11

Mathematics

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(JEE MAIN-2021)

If the distances from the origin to the centres of the three circles ${x^2} + {y^2} - 2{\lambda _i}\,x = {c^2},(i = 1,\,2,\,3)$ are in $G. P.$, then the lengths of the tangents drawn to them from any point on the circle ${x^2} + {y^2} = {c^2}$ are in

Solution

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