Suppose S_1 and S_2 are two unequal circles,A | vedclass

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(C) Let $R$ be a point on the direct common tangent $AB$ and $S$ be a point on the direct common tangent $CD$. The transverse common tangent $PQ$ inte

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$10$

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(C) Let $R$ be a point on the direct common tangent $AB$ and $S$ be a point on the direct common tangent $CD$. The transverse common tangent $PQ$ intersects $AB$ at $R$ and $CD$ at $S$.From point $R$,the tangents to the circle $S_2$ are $RA$ and $RQ$. Since tangents from an external point to a circle are equal in length,we have $RA = RQ$.Similarly,from point $S$,the tangents to the circle $S_2$ are $SP$ and $SD$. Thus,$SP = SD$.Also,from point $R$,the tangents to the circle $S_1$ are $RA$ and $RP$. Thus,$RA = RP$.From point $S$,the tangents to the circle $S_1$ are $SC$ and $SP$. Thus,$SC = SP$.We know that $AB$ is the length of the direct common tangent. Since $R$ lies on $AB$,$AB = AR + RB$. Since $RA = RQ$,we have $AB = RQ + RB$.For the transverse tangent segment $RS$,we have $RS = RP + PQ + QS$ is not correct; rather,$R$ and $S$ are points on the direct tangents. The length of the segment $RS$ of the transverse tangent between the two direct tangents is equal to the length of the direct tangent segment $AB$ (or $CD$).Therefore,$RS = AB = 10$.

Suppose $S_1$ and $S_2$ are two unequal circles,$AB$ and $CD$ are the direct common tangents to these circles. $A$ transverse common tangent $PQ$ cuts $AB$ in $R$ and $CD$ in $S$. If $AB=10$,then $RS$ is

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