Home
Class 12
MATHS
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

`((sqrt3)/(2),(3)/(2)), (sqrt3,0)`

B

`((sqrt3)/(2),(1)/(2)), (sqrt3,0)`

C

`((sqrt3)/(2),(3)/(2)), ((sqrt3)/(2),(1)/(2))`

D

`((3)/(2),(sqrt3)/(2)), ((sqrt3)/(2),(1)/(2))`

Text Solution

Verified by Experts

The correct Answer is:
A
Slope of line joining centre of circle to point D is
`tantheta=((3)/(2)-1)/((3sqrt3)/(2)-sqrt3)=(1)/(sqrt3)`
It makes an angle `30^(@)` with X-axis.
`therefore` Points E and F will make angle `150^(@) and -90^(@)` with X-axis.

`therefore` E and F are given by
`(x-sqrt3)/(cos150^(@))=(y-1)/(sin150^(@))=1`
`and (x-sqrt3)/(cos(-90^(@)))=(y-1)/(sin(-90^(@)))=1`
`therefore E=((sqrt3)/(2),(3)/(2)and F=(sqrt3,0)`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 3 Fill in the blanks|2 Videos

  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 3 Analytical & Descriptive Questions|3 Videos

  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 3 Passage Based Problems (passage 1)|3 Videos

  • BINOMIAL THEOREM

    IIT JEE PREVIOUS YEAR|Exercise Topic 2 Properties of Binomial Coefficent Objective Questions I (Only one correct option) (Analytical & Descriptive Questions )|8 Videos

  • COMPLEX NUMBERS

    IIT JEE PREVIOUS YEAR|Exercise TOPIC 5 DE-MOIVRES THEOREM,CUBE ROOTS AND nth ROOTS OF UNITY (INTEGER ANSWER TYPE QUESTION)|1 Videos

Similar Questions

Explore conceptually related problems

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

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

Knowledge Check

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

    A
    `((sqrt(3))/(2), (3)/(2)), (sqrt(3),0)`
    B
    `((sqrt(3))/(2), (1)/(2)), (sqrt(3),0)`
    C
    `((sqrt(3))/(2), (3)/(2)), ((sqrt(3))/(2),(1)/(2))`
    D
    `((3)/(2), (sqrt(3))/(2)), ((sqrt(3))/(2),(1)/(2))`

  • In a triangle PQR, PQ = 20 cm and PR = 6 cm, the side QR is :

    A
    equal to 14 cm
    B
    less than 14 cm
    C
    greater than 14 cm
    D
    none of these

  • PQRS is a quadrilateral having 3 4 5 6 points on PQ , QR, RS and SP respectively. The number of triangles with vertices on different sides is

    A
    220
    B
    270
    C
    282
    D
    none of these

    • Similar Questions

    Explore conceptually related problems

    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 and D, E, F respectively. The line PQ is given by the equation sqrt(3) +y-6=0 and the point D is ((sqrt(3))/(2), 3/2) . Equation of the sides QR, RP are : (A) y=2/sqrt(3) x + 1, y = 2/sqrt(3) x -1 (B) y= 1/sqrt(3) x, y=0 (C) y= sqrt(3)/2 x + 1, y = sqrt(3)/2 x-1 (D) y=sqrt(3)x, y=0

    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 and D, E, F respectively. The line PQ is given by the equation sqrt(3) +y-6=0 and the point D is ((sqrt(3))/(2), 3/2) . Point E and F are given by : (A) (sqrt(3)/2, 3/2), (sqrt(3), 0) (B) (sqrt(3)/2, 3/2), (sqrt(3)/2, 1/2) (C) (3/2, sqrt(3)/2), (sqrt(3)/2, 1/2)

    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 and D, E, F respectively. The line PQ is given by the equation sqrt(3) +y-6=0 and the point D is ((sqrt(3))/(2), 3/2) The equation of circle C is : (A) (x-2sqrt(3))^2 + (y-1)^2 = 1 (B) (x-2sqrt(3))^2 + (y+1/2)^2 = 1 (C) (x-sqrt(3))^2 + (y+1)^2 = 1 (D) (x-sqrt(3))^2 + (y-1)^2 =1

    Number of equilateral triangle with y=sqrt3(x-1)+2;y=- sqrt3x as two of its sides is

    The vertex P of an equilateral triangle /_PQR is at (2,3) and the equation of the opposite side QR is given by x+y=2. Find the possible equations of the side PQ.

    Home
    Profile