Question

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2+y^2=16$, by the lines, $x+y=n,n\in$ N, where N is the set of all natural numbers, is :

• A. 320
• B. 160
• C. 105
• D. 210

Answer 1

Answer: (D)

$p=\frac{n}{\sqrt{2}}\frac{n}{\sqrt{2}}<4\Rightarrow n=1,2,3,4,5.$
Length of chord $AB=2\sqrt{16-\frac{n^2}{2}}$
$=\sqrt{64-2n^2}=\ell \left(say\right)$
For $n=1,{\ell }^2=62$
$n=2,{\ell }^2=56$
$n=3,{\ell }^2=46$
$n=4,{\ell }^2=32$
$n=5,{\ell }^2=14$
$\therefore$ Required sum = $62+56+46+32+14=210$