Obtaining Centre and Radius of a Circle from General Equation of a Circle
The coordinat...
Question
The coordinates of the centre of a circle, whose radius is 2 units and which touches the line pair x2−y2−2x+1=0, are
A
(4,0)
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B
(1+2√2,0)
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C
(4,1)
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D
(1,2√2)
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Solution
The correct option is D(1,2√2) Line pair is (x−1)2−y2=0,
i.e., x+y−1=0,x−y−1=0
If the centre is on the x-axis,
let the centre be (α,0).
Then its distance from x+y−1=0 is ∣∣∣α−1√2∣∣∣=2(radius)
i.e., α=1±2√2 ∴ Centre may be (1+2√2,0),(1−2√2,0)
If the centre is on the line x=1,
let the centre be (1,β).
Then ∣∣∣1+β−1√2∣∣∣=2 ⇒β=±2√2 ∴ Centre may be (1,2√2),(1,−2√2)