Circles
Distance Between Two Chords
nta_pyq_2024_apr
Grade 11
Question:
Let $C$ be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x+y=2$ intersect the circle $C$ at the points $P$ and $Q$. Let $MN$ be a chord of $C$ of length 2 units and slope $-1$. Then a distance (in units) between the chord $PQ$ and the chord $MN$ is:
$3-\sqrt{2}$
$\sqrt{2}+1$
$\sqrt{2}-1$
$2-\sqrt{3}$
Step-by-Step Solution
Key Concept: Perpendicular distance from origin to $PQ$ ($x+y=2$): $d_{PQ}=|0+0-2|/\sqrt{2}=\sqrt{2}$. For $MN$ (length 2, slope $-1$): half-chord $=1$, $OA^2=10-1=9\Rightarrow OA=3$. Distance between chords $=OA\pm d_{PQ}=3\pm\sqrt{2}$.
$d_{PQ}=\sqrt{2}$, $OA=3$. Distance $=3-\sqrt{2}$.
Correct Answer: 1