Question
If the points of intersection of the ellipses x^ 2 +2 y^ 2 -6 x-12 y+23=0 and 4 x^ 2 +2 y^ 2 -20 x-12 y+35=0 lie on a circle of radius r and centre (a, b), then the value of a b+18 r^ 2 is
JEE Main
Mathematics
Circle
2026
JEE Main 2026 (23 January Shift 2)
JEE Main Mathematics Question (2026) — Solution
Options
- A. 52
- B. 53
- C. 51
- D. 55
Answer
D. 55
Step-by-step solution
The circle through intersection of the ellipses is S_1 + S_2 = 0. (1+4 )x^2 + (2+2 )y^2 - (6+20 )x - (12+12 )y + (23+35 ) = 0. For a circle: 1+4 = 2+2 = 1 2 . Circle: 3x^2 + 3y^2 - 16x - 18y + 81 2 = 0 x^2 + y^2 - 16 3 x - 6y + 27 2 = 0. Centre (a,b) = ( 8 3 , 3 ), r^2 = 64 9 + 9 - 27 2 = 47 18 . ab + 18r^2 = 8 3 3 + 18 47 18 = 8 + 47 = 55.
Related: Mathematics — Circle · All PYQ Banks