Question

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

Answer 1

let the radius of on circle be R
let the radius of other circle be R
according to question,
22/7(R^2+r^2)=130
R^2+r^2=130×7/22
R^2+r^2=455/11


R+r=14
R=14-r
put these value
(14-r)^2+r^2=455/11
simplify it u will get answer

Answer 2

Since the given circles touch externally, we have

Sum of their radii = distance between their centres= 14cm.

$\:\:$

${\bold { \underline{\small{ Let \:the \:radii\: of \:the \:given \:circles \:be -}}}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\normalsize{\underline{\mathcal{\green{x\:cm\:\:\ and }}}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\normalsize{\underline{\mathcal{\red{(14 - X) \:\:cm }}}}$

$\:\:$

$\sf Sum\: of \:their\: areas \:= [πx²+ π(14-x)²]cm².$

$\:\:$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\sf \therefore\:\:\:πx²+π(14-x)²=130π$

$\:\:$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\hookrightarrow\:$$\sf x²+(14-x)²=130$

$\:\:$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\hookrightarrow\:$$\sf 2x²-28x+66=0$

$\:\:$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\hookrightarrow\:$$\sf x²-14x+33=0$

$\:\:$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\hookrightarrow\:$$\sf (x-11)(x-3)=0$

$\:\:$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\hookrightarrow\:$$\sf x-11=0\:\:\:or\:\:\:x-3=0$

$\:\:$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\underline{\boxed{\sf{x=\frak{\purple{11}}}}}$ $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\underline{\boxed{\sf{x=\frak{\orange{3}}}}}$

$\:\:$

Now, put the value of x

$\sf\bold x = 11$

• $\sf (14-x)=(14-11)=3$

$\sf\bold x = 3$

• $\sf(14-x)=(14-3)=11.$

$\:\:$

Hence the radii of the circles are 11cm and 3cm.

$\:\:$

$\:\:$

${\small{\color{green}\star{\mathfrak{\color{teal} {\underline{\underline{Related \:terms\: and \:formulas :}}}}}}}$

$\:\:$

CIRCLE

• A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point always remains the same.

The fixed point is called the center and

The given constant distance is known as the radius of the circle.

Segment of a Circle: The portion (or part) of a circular region enclosed between a chord and the corresponding arc is called a segment of the circle.

Sector of a Circle: The portion (or part) of the circular region enclosed by the two radii and the corresponding arc is called a sector of the circle.

$\:\:$

$\sf Area\: of \:circle= 2πr\:\:\:\:or\:\:\:\:\bf\dfrac{πd²}{4}$|where, 'r' = radius|

$\sf Area\: of \:semicircle = [tex]\bf\dfrac{πr²}{2}$

$\sf Radius\: of \:circle = \displaystyle \sqrt {\frac{Area}{π}}$