Question:
Choose the incorrect statement about the two circles whose equations are given below :
\(x^2 + y^2 - 10x - 10y + 41 = 0\) and
\(x^2 + y^2 - 16x - 10y + 80 = 0\)
Choose the incorrect statement about the two circles whose equations are given below :
\(x^2 + y^2 - 10x - 10y + 41 = 0\) and
\(x^2 + y^2 - 16x - 10y + 80 = 0\)
\(x^2 + y^2 - 10x - 10y + 41 = 0\) and
\(x^2 + y^2 - 16x - 10y + 80 = 0\)
Show Hint
Find both centres and radii, then compare the distance between centres with the radii.
Updated On: Nov 4, 2026
- Distance between two centres is the average of radii of both the circles.
- Both circles pass through the centre of each other.
- Circles have two intersection points.
- Both circle's centers lie inside region of one another.
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The Correct Option is D
Solution and Explanation
Circle 1: centre \(C_1(5,5)\), \(r_1 = \sqrt{25 + 25 - 41} = 3\).
Circle 2: centre \(C_2(8,5)\), \(r_2 = \sqrt{64 + 25 - 80} = 3\).
\(C_1C_2 = 3\).
(A) Average of radii \(= \frac{3+3}{2} = 3 = C_1C_2\): true.
(B) \(C_1C_2 = 3 = r_1 = r_2\), so each circle passes through the other's centre: true.
(C) \(|r_1 - r_2| = 0 < C_1C_2 = 3 < r_1 + r_2 = 6\), so there are two intersection points: true.
(D) The centre of one circle lies at distance exactly 3 = radius from the other centre, i.e. ON the other circle, not inside it: false.
Answer: (4) Both circle's centers lie inside region of one another.
Circle 2: centre \(C_2(8,5)\), \(r_2 = \sqrt{64 + 25 - 80} = 3\).
\(C_1C_2 = 3\).
(A) Average of radii \(= \frac{3+3}{2} = 3 = C_1C_2\): true.
(B) \(C_1C_2 = 3 = r_1 = r_2\), so each circle passes through the other's centre: true.
(C) \(|r_1 - r_2| = 0 < C_1C_2 = 3 < r_1 + r_2 = 6\), so there are two intersection points: true.
(D) The centre of one circle lies at distance exactly 3 = radius from the other centre, i.e. ON the other circle, not inside it: false.
Answer: (4) Both circle's centers lie inside region of one another.
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