Let C be a circle with radius units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 units and slope –1. Then a distance (in units) between the chord PQ and the chord MN is

Question

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 units and slope –1. Then a distance (in units) between the chord PQ and the chord MN is [2024]

Smart Achievers · Accepted Answer

(1)

Distance of centre from chord $P Q =$ $| \frac{0 + 0 - 2}{\sqrt{1^{2} + 1^{2}}} |$ $= \frac{2}{\sqrt{2}} = \sqrt{2}$

Let distance of centre from chord MN = p, then length of chord = $2 \sqrt{10 - p^{2}}$

$\Rightarrow 2 \sqrt{10 - p^{2}} = 2$ ( $∵$ Given)

$\Rightarrow 10 - p^{2} = 1 \Rightarrow p^{2} = 9 \Rightarrow p = 3$

$\Rightarrow$ Distance between chord PQ and MN = $3 - \sqrt{2}$ .

Solution

Q.
Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 units and slope –1. Then a distance (in units) between the chord PQ and the chord MN is [2024]

1 $3 - \sqrt{2}$

2 $2 - \sqrt{3}$

3 $\sqrt{2} + 1$

4 $\sqrt{2} - 1$

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Solution

Q. Let C be a circle with radius 10 units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 units and slope –1. Then a distance (in units) between the chord PQ and the chord MN is [2024]

1 3–2

2 2–3

3 2+1

4 2–1