Two circles touch externally. The sum of their areas is 130π sq.cm. and the distance between their centre is 14cm. Find the radii of the circles ____.

Question

Two circles touch externally. The sum of their areas is 130π sq.cm. and the distance between their centre is 14cm. Find the radii of the circles ____.

Accepted Answer

Let us consider two circles with r1 and r2 as their radii and the larger be having the radii as r1
Question Image

Now we are given that the distance between the centres of the circles is 14cm that is-

r_{1}

+

r_{2}

=14cm

So we get the needed relation between the r1 and r2

Now also the sum of the areas of the two circles is 130πsq.cm. let us suppose the areas of the two circles be A1 and A2 so the another relation that is between the areas of the two circles is as- A1+A2=130.π.sq.cm
Now the area of any circle with radius r is as-

π r^{2}

Now the relation of the areas become

{π r_{1}}^{2} + {π r_{2}}^{2} = 130 π

sq.cm
Now taking like terms common and simplifying the terms we get –

{π {(r}_{1}}^{2} + {r_{2}}^{2}) = 130 π

sq.cm----------------1
Now using the relation between the

r_{1}

and

r_{2}

we will find the value of

r_{2}

in the form of the

r_{1}

that is -
Now putting the value of the

r_{2}

from the above equation in the equation 1
that is

{π r_{1}}^{2} + {(14 - {r_{1}}^{2})}^{2} = 130 π

sq.cm
Now removing π from both sides the above equation becomes –
⇒

{r_{1}}^{2} + 196 + {r_{1}}^{2} - 28 r_{1} = 130

⇒2

{r_{1}}^{2}

−

28 r_{1}

+66=0
⇒

{r_{1}}^{2} - 14 r_{1}

+33=0
⇒

{r_{1}}^{2} - 11 r_{1} - 3 r_{1} + 33 = 0

⇒(

r_{1}

−11)(

r_{1}

−3)=0
Now from the above equation we get the (

r_{1}

−11) = 0 also (

r_{1}

−3) = 0
Now

r_{1}

=11,3 now putting

r_{1}

in the equation

r_{1} + r_{2} = 14

cm we get

r_{2}

=11,3
As depicted in the figure

r_{1}

and assumed earlier being the larger of the two

r_{1}

=11cm and

r_{2}

=3 cm so from the above given options the option which matches the resultant is the option 1 that is 1) 11 cm and 3 cm. So the correct option is 1

Two circles touch externally. The sum of their areas is 130π sq.cm. and the distance between their centre is 14cm. Find the radii of the circles ____.

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