Circles
Common Chord of Ellipses — Circle through Intersection
nta_pyq_2026_jan
Grade 11
Question:
If the points of intersection of the ellipses $x^2+2y^2-6x-12y+23=0$ and $4x^2+2y^2-20x-12y+35=0$ lie on a circle of radius $r$ and centre $(a,b)$, then the value of $ab+18r^2$ is
Step-by-Step Solution
Key Concept: Family: $S_1+\lambda S_2=0$. For a circle: coefficients of $x^2$ and $y^2$ must be equal. $(1+4\lambda)=(2+2\lambda)\Rightarrow\lambda=1/2$. Circle: $3x^2+3y^2-16x-18y+81/2=0$.
Centre $(8/3,3)$, $r^2=47/18$. $ab+18r^2=55$.
Correct Answer: 4