If two circles with radii 8 and 3 respectively find the distance between their centres

Two circles of radii 8 cm and 3 cm have a direct common tangent of length 10 cm. Find the distance between their centers, up to two places of decimal.

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Let us consider that ${O_1}$ and ${O_2}$ be the center of circle X and circle Y, respectively, as shown in the figure above. We have,Radius of circle X i.e. ${O_1}A$ = 8 cm andRadius of circle Y i.e. ${O_2}A$ = 3 cm. Also, the distance between the centers of both the circles is 13 cm$({O_1}{O_2})$.We have the common tangent as AB on both circles. We have done the construction from ${O_2}$ to ${O_1}A$ to intersect at P as shown.We can get the value of ${O_1}P$ as mentioned below,\[ \Rightarrow {O_1}P = {O_1}A - {O_2}B = 8 - 3 = 5cm.\]In $\vartriangle P{O_1}{O_2}$,$ \Rightarrow P{O_2} = \sqrt {{{({O_1}{O_2})}^2} - {{({O_1}P)}^2}} $ (Using Pythagoras Theorem)$ \Rightarrow P{O_2} = \sqrt {{{(13)}^2} - {{(5)}^2}} $$ \Rightarrow P{O_2} = 12cm$. This is the same distance as that of common tangent AB.Hence, option (C) is correct. Note- Here we have not directly calculated the length of the common tangent. We have first calculated the length of the line which is equal to it and parallel to it. We have used Pythagoras theorem as mentioned below:

$ \Rightarrow Hypotenuse = \sqrt {{{(Perpendicular)}^2} + {{(Base)}^2}} $.

If two circles with radii 8 cm and 3 cm respectively touch externally, then find the distance between their centres.

`"l"("C"_1"C"_2) = "r"_1 + "r"_2` = 8 + 3 = 11

If two circles touches externally then distance between their centres is equal to sum of the radii.

Concept: Touching Circles

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If two circles with radii 8 cm and 3 cm, respectively, touch internally, then find the distance between their centers.

If two circles touch internally, then distance between their centers is the difference of their radii.

The distance between their centers = 8 – 3 = 5 cm

Concept: Touching Circles

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