Circles
Chord as Diameter — Farthest Chord from Centre
nta_pyq_2026_jan
Grade 11
Question:
Let $y=x$ be the equation of a chord of the circle $C_1$ (in the closed half-plane $x\geq0$) of diameter 10 passing through the origin. Let $C_2$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_2$, which passes through the point $(2,3)$ and is farthest from the center of $C_2$, is $x+ay+b=0$, then $a-b$ is equal to
Step-by-Step Solution
Key Concept: $C_1$ has diameter 10, passes through origin with centre $(5,0)$ (in $x\geq0$ half-plane). Chord $y=x$ meets $C_1$ at $(0,0)$ and $(5,5)$. $C_2$ has centre $(5/2,5/2)$ and radius $5\sqrt{2}/2$.
Chord of $C_2$: $x-y+1=0$. $a=-1,b=1$. $a-b=-2$.
Correct Answer: 1