JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Circle

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

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

and

x^{2} + y^{2} - 16 x - 10 y + 80 = 0

(A)Distance between two centres is the average of radii of both the circles.
(B)Both circles' centres lie inside region of one another.✓ Correct
(C)Circles have two intersection points.
(D)Both circles pass through the centre of each other.

Answer: (B)

Solution

Completing the square, the circles are

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

with centre

C_{1} (5, 5)

and radius

3

, and

(x - 8)^{2} + (y - 5)^{2} = 9

with centre

C_{2} (8, 5)

and radius

3

. The distance between the centres is

C_{1} C_{2} = 3

, which equals the average of the radii, so that statement holds. Since

∣ r_{1} - r_{2} ∣ = 0 < 3 < r_{1} + r_{2} = 6

, the circles cut at two points, so that statement holds as well. And

C_{1} C_{2} = r_{1} = r_{2}

places each centre exactly on the other circle, so each circle passes through the centre of the other, yet neither centre is interior to the other circle. The incorrect statement is that both centres lie inside the region of one another.

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Single correct option

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JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Circle

Solution

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