Question

Two lisches touch externally. The sum of
their areas 13πsqcm
distance between their centres is 14cm. Find
the radii of each circle
and find the​

Answer 1

Step-by-step explanation:

If two circles touch externally, then the distance between their centers is equal to the sum of their radii. Let the radii of the two circles be r

1

cm and r

2

cm respectively.

Let C

1

and C

2

be the centres of the given circles. Then,

C

1

C

2

=r

1

+r

2

[∵C

1

C

2

=14cm (given)]

⇒14=r

1

+r

2

⇒r

1

+r

2

=14 ....(i)

It is given that the sum of the areas of two circles is equal to 130πcm

2

.

∴π(r

1

)

2

+π(r

2

)

2

=130π

⇒(r

2

)

2

+(r

2

)2=130 ...(ii)

Now, (r

1

+r

2

)

2

=(r

1

)

2

+(r

2

)

2

+2r

1

r

2

⇒142=130+2r

1

r

2

[Using (i) and (ii)]

⇒196−130=2r

1

r

2

⇒r

1

r

2

=33 ....(iii)

Now,

(r

1

−r

2

)

2

=(r

1

)

2

+(r

2

)

2

−2r

1

r

2

⇒(r

1

−r

2

)

2

=130−2×33 [Using (ii) and (iii)]

⇒(r

1

−r

2

)

2

=64

⇒r

1

−r

2

=8 ....(iv)

Solving (i) and (iv), we get r

1

=11cm and r

2

=3cm. Hence, the radii of the two circles are 11cm and 3cm.