Let a circle $C$ pass through the points $(4, 2)$ and $(0, 2)$, and its centre lie on $3x + 2y + 2 = 0$. Then the length of the chord, of the circle $C$, whose mid-point is $(1, 2)$, is
Step-by-Step Solution
Key Concept: Parameterise the centre on $3x+2y+2=0$, use equal distance from both given points to find the centre and radius, then apply chord length $= 2\sqrt{R^2 - d^2}$ where $d$ is the distance from the centre to the given midpoint $(1,2)$.
Parameterise the centre as $(-2a, 3a-1)$ on $3x+2y+2=0$. Equidistance from $(4,2)$ and $(0,2)$: $$(4+2a)^2+9(a-1)^2=4a^2+9(a-1)^2 \implies 16a+16=0 \implies a=-1.$$ Centre $=(2,-4)$, $R^2=(4-2)^2+(2+4)^2=4+36=40$. Distance from $(2,-4)$ to midpoint $(1,2)$: $d^2=1+36=37$. Chord length $=2\sqrt{40-37}=2\sqrt{3}$.
<div class="key-concept"><strong>Key Concept:</strong> Parameterise the centre on $3x+2y+2=0$, use equal distance from both given points to find the centre and radius, then apply chord length $= 2\sqrt{R^2 - d^2}$ where $d$ is the distance from the centre to the given midpoint $(1,2)$.</div>
<div class="trap-box"><strong>Trap:</strong> Confusing the given midpoint $(1,2)$ with a point on the circle and trying to substitute it into the circle equation directly, bypassing the perpendicular-distance step.</div>
Correct Answer: 3