Home
Class 12
MATHS
inside the circles x^2+y^2=1 there are t...

inside the circles `x^2+y^2=1` there are three circles of equal radius `a` tangent to each other and to `s` the value of `a` equals to

A

`sqrt(2) (sqrt(2)-1)`

B

`sqrt(3)(2-sqrt(3))`

C

`sqrt(2)(2-sqrt(3))`

D

`sqrt(3)(sqrt(3)-1)`

Text Solution

Verified by Experts

The correct Answer is:
B

We have `AC = 2a`
and `OA = OC =1 -a` (since circles are touching internally)
Since `DeltaABC`is equilateral, we have
`:. /_AOC = 120^(@)`
Using cosine rule, we have
`cos 120^(@) = ((1-a)^(2)+(1-a)^(2)-(2a)^(2))/(2(1-a)(1-a))`
`rArr -(1)/(2) =(-a^(2)-2a+1)/((1-a)^(2))`
`rArr a^(2)-2a +1 =2a^(2) +4a -2`
`rArr a^(2)+6a -3 =0`
`rArr a = (-6+sqrt(48))/(2) = sqrt(3) (2-sqrt(3))`
Promotional Banner

Topper's Solved these Questions

  • CIRCLES

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|9 Videos

  • CIRCLES

    CENGAGE PUBLICATION|Exercise Comprehension Type|8 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise For problems 3 and 4|2 Videos

  • COMPLEX NUMBERS

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

Similar Questions

Explore conceptually related problems

Prove that the centres of three equal circles touchinhg each other are the certices of an equilateral triangle.

Consider the circles x^2+(y-1)^2=9,(x-1)^2+y^2=25. They are such that a.these circles touch each other b.one of these circles lies entirely inside the other c.each of these circles lies outside the other d.they intersect at two points.

A tangent at a point on the circle x^2+y^2=a^2 intersects a concentric circle C at two points Pa n dQ . The tangents to the circle X at Pa n dQ meet at a point on the circle x^2+y^2=b^2dot Then the equation of the circle is (a) x^2+y^2=a b (b) x^2+y^2=(a-b)^2 (c) x^2+y^2=(a+b)^2 (d) x^2+y^2=a^2+b^2

If tangent at (1, 2) to the circle C_1: x^2+y^2= 5 intersects the circle C_2: x^2 + y^2 = 9 at A and B and tangents at A and B to the second circle meet at point C, then the co- ordinates of C are given by

Find the locus of the centers of the circles x^2+y^2-2ax-2b y+2=0 , where a and b are parameters, if the tangents from the origin to each of the circles are orthogonal.

With one focus of the hyperbola x^2/9-y^2/16=1 as the centre, a circle is drawn which is tangent to the hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is

The equation of the locus of the middle point of a chord of the circle x^2+y^2=2(x+y) such that the pair of lines joining the origin to the point of intersection of the chord and the circle are equally inclined to the x-axis is (a) x+y=2 (b) x-y=2 (c) 2x-y=1 (d) none of these

In the adjoining figure, O is the centre of a circle and AB is one of its diameters. The legth of the chrod CD is equal to the radius of the circle. When AC and BD are extended, they intersect each other at a point P. find the value of angle APB

Construct a circle of radius 2.5 cm. Take any point on the circle and draw a tangent to the circle at that point.

Draw a pair of tangents to a circle of radius 5cm which are inclined to each other at an angle 60^(@) .