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A circle C of radius 1 is inscribed in a...

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation `sqrt3 x+ y -6 = 0` and the point D is (3 sqrt3/2, 3/2). Further, it is given that the origin and the centre of C are on the same side of the line PQ. (1)The equation of circle C is (2)Points E and F are given by (3)Equation of the sides QR, RP are

A

`(x-2sqrt3)^(2)+(x-1)^(2)=1`

B

`(x-2sqrt3)^(2)+(y-(1)/(2))^(2)=1`

C

`(x-sqrt3)^(2)+(y+1)^(2) = 1`

D

`(x-sqrt3)^(2)+(y-1)^(2) = 1`

Text Solution

Verified by Experts

The correct Answer is:
D
Let centre of circle C be (h,k).
Then, `|(sqrt3h+k-6)/(sqrt(3+1))|=1`
`rArr sqrt3h+k-6=2,-2`
`rArr sqrt3h+k=4" "...(i)`
[rejecting 2 because origin and centre of C are on the same side of PQ]
The point `(sqrt3,1)` satisfies Eq. (i).
`therefore` Equation of circle C is `(x-sqrt3)^(2)+(y-1)^(2)=1`.
Clearly, point E and F satisfy the equation given in option (d).
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