JEE Main 2021 · 17 March · morning · Single correct · +4 / −1
Choose the incorrect statement about the two circles whose equations are given below :
x² + y² - 10x - 10y + 41 = 0 and
x² + y² - 16x - 10y + 80 = 0
- (a) Distance between two centres is the average of radii of both the circles.
- (b) Both circles pass through the centre of each other.
- (c) Circles have two intersection points.
- (d) Both circle's centers lie inside region of one another.
Answer and solution
Answer (d) Both circle's centers lie inside region of one another.
Solution
Maths sir · 3 steps
Find both centres and radii first; then every option is a comparison of d with r.
- Centres and radii. For x² + y² + 2gx + 2fy + c = 0, the centre is (-g, -f) and r² = g² + f² - c. The first circle has centre C₁(5, 5) and r₁² = 25 + 25 - 41 = 9, so r₁ = 3. The second has centre C₂(8, 5) and r₂² = 64 + 25 - 80 = 9, so r₂ = 3.
- Distance between centres. Both centres have y = 5, so d = C₁C₂ = 8 - 5 = 3. The distance between the centres equals each radius.
Option Check Verdict (a) (r₁ + r₂)/2 = (3 + 3)/2 = 3 = d true (b) d = r₁ = r₂ = 3, so each centre is at distance r from the other centre true (c) |r₁ - r₂| = 0 < d = 3 < r₁ + r₂ = 6 true (d) Inside would need d < r, but d = r, so each centre lies on the other circle false - So. Putting C₁(5, 5) into the second equation gives 25 + 25 - 80 - 50 + 80 = 0, so C₁ is on the second circle, not inside it. The incorrect statement is (d).
The idea to keep. To place a point relative to a circle, compare its distance from the centre with r. Less than r means inside, equal to r means on the circle, more than r means outside.
Answer from the NTA final answer key. Solution by PrepWiser.