Question

The sum of the square of the lengths of the chords intercepted by the lines $x+y=n,nϵN$on the circle $x^2+y^2=4$

• A. $106$
• B. $2024$
• C. $22$
• D. $18$

Answer 1

Answer: (C)

$22$

If AB is the chord $x+y=n$, then OM $=\frac{n}{\sqrt{2}}$ ...By perpendicular distance formula
Thus $(AB{)}^2=(2AM{)}^2$ ....Perpendicular drawn from center to the chord bisects the chord

In $△OAM$,

$O{A}^2=O{M}^2+A{M}^2$ ....By Pythagoras theorem

$\Rightarrow A{M}^2=O{A}^2-O{M}^2$

$\Rightarrow 4(AM{)}^2=4[2^2-\frac{n^2}{2}]=2(8-n^2)$
Hence the required sum $=2[8-1^2+8-2^2]=2(16-5)=22$