two circles touch Each Other externally the sum of the areas is 149 Pi cm and the distance between the centres is 17 cm find the radius of the circle
Answers
Answer:
Let the radius of the two circles are a and b cm
The circles touch each other externally.
So, the distance between their centers =a+b
a+b=14 ................... (1)
Sum of their areas =πa
2
+πb
2
πa
2
+πb
2
=130π
a
2
+b
2
=130 .................... (2)
From equation (1),
b=14−a ............. (3)
From (2) and (3),
a
2
+(14−a)
2
=130
a
2
+a
2
+196−28a=130
2a
2
+66−28a=0
a
2
+33−14a=0
a
2
−11a−3a+33=0
(a−11)(a−3)=0
a=3,11
If a=3, then b=14−3=11
If a=11, then b=14−11=3
So, the radius of the two circles are 3 and 11 cm
And their diameters are 6 and 22 cm.
Given :
- Sum of area of two circles is 149π cm².
- Distance between the centre of two circles is 17 cm.
To find :
- Radius of the two circles?
Solution :
☯ Let radius of two circles be r and R cm.
★ According to the Question:
- The circles touch each other externally.
So,
The distance between their centers is,
⇒ r + R = 17
⇒ R = 17 - r⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀❬ eq. (1) ❭
Also,
- Sum of the areas of two circle is 149 π cm².
⇒ πr² + πR² = 149π
⇒ π(r² + R²) = 149π
⇒ r² + R² = 149 ⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀❬ eq. (2) ❭
⠀⠀━━━━━━━━━━━━━━━━━━━━━
Now, Putting eq. (1) in eq. (2),
⇒ r² + (17 - r)² = 149
⇒ r² + 289 + r² - 34r = 149
⇒ 2r² - 34r + 289 - 149 = 0
⇒ 2r² - 34r + 140 = 0
Splitting middle term,
⇒ 2(r² - 17r + 70) = 0
⇒ r² - 17r + 70 = 0
⇒ r² - 10r - 7r + 70 = 0
⇒ r(r - 10) - 7(r - 10) = 0
⇒ (r - 10)(r - 7) = 0
either (r - 10) or (r - 7) is equal to 0.
Therefore,
⇒ r = 10, 7
Now, From eq. (1),
- If r = 10, then R = 17 - 10 = 7
- If r = 7, then R = 17 - 7 = 10
∴ Hence, the radius of two circles are 7 and 10 cm.