Circles
Circle
star_batch_jee_advanced_2025
Grade None
Question:
If the circle passing through the distinct points $(a,t), (t,a)$ and $(r,t)$ for all values of '$r$' passes through a fixed point then
possible values of $a$ is $4$
possible values of $a$ is $8$
possible values of $a$ is $12$
possible values of $a$ is $15$
Step-by-Step Solution
Key Concept: When three points with a specific symmetric structure lie on a circle, the resulting circle always maintains a tangency condition with a fixed line at a fixed point.
Given the circle equation $x^2 + y^2 + 2gx + 2fy + c = 0$ with points $(a,t)$, $(t,a)$, and $(t,t)$ lying on it, substitute each point to obtain three equations. After algebraic manipulation, the circle $x^2 + y^2 - ax - ay = 0$ always touches the line $x + y - 2a = 0$ at the point $(a, a)$, independent of the parameter values.
Correct Answer: 1,2,3,4