Choose the correct statement about two circles whose equations are given below: \[ \begin{aligned} & x^{2}+y^{2}-10 x-10 y+41=0 \\ & x^{2}+y^{2}-22 x-10 y+137=0 \end{aligned} \]

(C) circles have only one meeting point

(A) circles have no meeting point

(B) circles have two meeting points

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Circle

Choose the correct statement about two circles whose equations are given below:

x^{2} + y^{2} - 10 x - 10 y + 41 = 0 x^{2} + y^{2} - 22 x - 10 y + 137 = 0

(A)circles have no meeting point
(B)circles have two meeting points
(C)circles have only one meeting point✓ Correct
(D)circles have same centre

Answer: (C)

Solution

Completing squares gives

(x - 5)^{2} + (y - 5)^{2} = 9, (x - 11)^{2} + (y - 5)^{2} = 9

so the centres are

(5, 5)

and

(11, 5)

, each with radius

3

. The distance between the centres is

d = (11 - 5)^{2} + (5 - 5)^{2} = 6

which equals

r_{1} + r_{2} = 3 + 3

. The circles touch externally, so they have only one meeting point.

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Single correct option

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