Circles
Circle Tangent to Axes Through a Point
nta_pyq_2024_apr
Grade 11
Question:
Let a circle $C$ of radius 1 and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $C$ from the point $(5,5)$ is:
$2\sqrt{2}$
$4\sqrt{2}$
4
5
Step-by-Step Solution
Key Concept: Lines through $(3,2)$ parallel to axes are $x=3$ and $y=2$. Circle of radius 1 tangent to both: centre at $(2,1)$ (closer to origin). Verify: distance from $(2,1)$ to $x=3$ is 1 ✓, to $y=2$ is 1 ✓.
Centre $(2,1)$, radius 1. $QC=\sqrt{(5-2)^2+(5-1)^2}=5$. Shortest distance $=5-1=4$.
Correct Answer: 3