SATHEE: Circle 5 Question 2

Circle 5 Question 2

2. 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

(2019, Main, 8 April I)

(a) 320

(b) 105

(d) 210

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Answer:

Correct Answer: 2. (d)

Solution:

Given equation of line is $x + y = n, n \in N$

and equation of circle is $x^{2} + y^{2} = 16$

Now, for intercept, made by circle (ii) with line (i)

$\Rightarrow \begin{matrix} d < 4 \\ \frac{n}{\sqrt{2}} < 4 \end{matrix}$

$[∵ d =$ perpendicular distance from $(0, 0)$ to the line $x + y = n$ and it equal to $\frac{| 0 + 0 - n |}{\sqrt{1^{2} + 1^{2}}} = \frac{n}{\sqrt{2}}$

$\Rightarrow n < 4 \sqrt{2}$

$∵ n \in N$ , so $n = 1, 2, 3, 4, 5$

Clearly, length of chord $A B = 2 \sqrt{4^{2} - d^{2}}$

$= 2 \sqrt{16 - \frac{n^{2}}{2}} ∵ d = \frac{n}{\sqrt{2}}$

$∴$ Sum of square of all possible lengths of chords (for $n = 1, 2, 3, 4, 5$ )

$\begin{aligned} = 4 (16 \times 5) - \frac{1}{2} (1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2}) \\ = 320 - 2 \frac{5 (6) (11)}{6} = 320 - 110 = 210 \end{aligned}$

Circle 5 Question 2

2. 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

Answer:

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Circle 5 Question 2

2. The sum of the squares of the lengths of the chords intercepted on the circle, x2+y2=16, by the lines, x+y=n,n∈N, where N is the set of all natural numbers, is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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2. 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