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Question: The equation of the circle of minimum radius which contains the three circles x² + y² – 4y – 5 = 0 ...

The equation of the circle of minimum radius which contains the three circles

x² + y² – 4y – 5 = 0 … (1) x² + y² + 12x + 4y + 31 = 0 … (2) and x² + y² + 6x + 12y + 36 = 0 … (3) is

A

(x−3118)2+(y−2312)2=value(x - \frac { 31 } { 18 })^{ 2 } + (y - \frac { 23 } { 12 })^{ 2 } = \text{value}

B

(x+2312)2+(y+3118)2=(3+536949)2(x + \frac { 23 } { 12 })^{ 2 } + (y + \frac { 31 } { 18 })^{ 2 } = \left( 3 + \frac { 5 } { 36 } \sqrt { 949 } \right)^{ 2 }

C

(x+3118)2+value=(3+536949)2(x + \frac { 31 } { 18 })^{ 2 } + \text{value} = \left( 3 + \frac { 5 } { 36 } \sqrt { 949 } \right)^{ 2 }

D

None of these

Answer

(x+3118)2+value=(3+536949)2(x + \frac { 31 } { 18 })^{ 2 } + \text{value} = \left( 3 + \frac { 5 } { 36 } \sqrt { 949 } \right)^{ 2 }

Explanation

Solution

The coordinates of the centres and radii of three given circles are as given below :

Centre Radius

Circle (1) C1(0, 2) r1 = 3

Circle (2) C2(–6, –2) r2 = 3

Circle (3) C3 (–3, –6) r3 = 3

Let C (h, k) be the centre of the circle passing through the centres of the circles (1), (2) and (3). Then,

CC1 = CC2 = CC3

Ž CC12 = CC22 = CC32

Ž (h – 0)2 + (k – 2)2 = (h + 6)2 + (k + 2)2

= (h + 3)2 + (k + 6)2

Ž –4k + 4 = 12h + 4k + 40 = 6h + 12k + 45

Ž 12h + 8k + 36 = 0 and 6h – 8k – 5 = 0

Ž 3h + 2k + 9 = 0 and 6h – 8k – 5 = 0

Ž h = −3118\frac { - 31 } { 18 }, k = −2312\frac { - 23 } { 12 }

\ CC1 = (0+3118)2+(2+2312)2\sqrt { \left( 0 + \frac { 31 } { 18 } \right) ^ { 2 } + \left( 2 + \frac { 23 } { 12 } \right) ^ { 2 } } = 536\frac { 5 } { 36 } 949\sqrt { 949 }

Now, CP = CC1 + C1P  CP = (536949+3)\left( \frac { 5 } { 36 } \sqrt { 949 } + 3 \right)

Thus, required circle has its centre at (−3118,−2312)\left( - \frac { 31 } { 18 } , - \frac { 23 } { 12 } \right) and radius = CP =(536949+3)\left( \frac { 5 } { 36 } \sqrt { 949 } + 3 \right).

Hence, its equation is

(3+536949)2\left( 3 + \frac { 5 } { 36 } \sqrt { 949 } \right) ^ { 2 }.

Hence (3) is the correct answer.