JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Circle

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✓ Correct

Answer: (D)

Solution

The circle

x^{2} + y^{2} = 16

has centre at the origin and radius

4

, and the line

x + y = n

lies a distance

p = \frac{n}{2}

from the centre. It cuts the circle only while

p < 4

, that is

n < 42 \approx 5.66

, so

n

runs over

1, 2, 3, 4, 5

. Each chord has length squared

ℓ^{2} = 4 (r^{2} - p^{2}) = 4 (16 - \frac{n ^{2}}{2}) = 64 - 2 n^{2}

, giving

62, 56, 46, 32, 14

for the five values, whose sum is

62 + 56 + 46 + 32 + 14 = 210

JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Single correct option

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