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Definition of Circle

The centres o...

Question

The centres of two circles $C_{1}$ and $C_{2}$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_{1}$ and $C_{2}$ and $C$ be a circle touching circles $C_{1}$ and $C_{2}$ extemally. If a common tangent to $C_{1}$ and $C$ passing through $P$ is also a common tangent to $C_{2}$ and $C$ , then the radius ofthe circle $C$ is

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Solution

$O, P = 3, O, A = 1$
$A P = \sqrt{O, P^{2} - O, A^{2}} = \sqrt{3^{2} - 1} = \sqrt{8} = 2 \sqrt{2}$
$O P^{2} - r^{2} + P A^{2} = r^{2} + (2 \sqrt{2})^{2}$
$O P^{2} = 8 + r^{2} . . . . . . . . . . . . . . . . . . . . . . (1)$
${O O}_{2}^{2} = O P^{2} + {P O}_{2}^{2}$
$(r + 1)^{2} = 8 + r^{2} + 3^{2}$
$r^{2} = 2 r + 1 = r^{2} + 17$
$2 r = 16$
$r = 8$

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Similar questions

Q. The centres of two circles

C_{1}

C_{2}

each of unit radius are at a distance of

6

units from each other. Let

P

be the mid point of the line segment joining the centres of

C_{1}

C_{2}

C

be a circle touching

C_{1}

C_{2}

externally. If a common tangent to

C_{1},

C

passing through

P

is also a common tangent to

C_{2}

C

, then the radius of the circle

C

C_{1}

C_{2}

are circles of unit radius with centres at

(0, 0)

(1, 0)

C_{3}

is a circle of unit radius, passes through the centres of the circles

C_{1}

C_{2}

and have its centre above x-axis. Equation of the common tangent to

C_{1}

C_{3}

which does not pass through

C_{2}

C_{1} and C_{2}

are circles of unit radius with centres at

(0, 0) and (1, 0)

C_{3}

is a circle of radius

r (r \in N)

. It passes through the centres of the circles

C_{1} and C_{2}

and has its center above the

x -

axis. If the common tangent of

C_{1} and C_{3}

\sqrt{3} x - y + c = 0

. Then maximum value of

c + r =

C_{1}

2

C_{2}

3

, and the distance between the centers of

C_{1}

C_{2}

7

. If two lines, one tangent to both circles and other passing through the center of both circles, intersect at point

P

which lies between centers of

C_{1}

C_{2}

, the the distance between

P

and the center of

C_{1}

C_{1}

C_{2}

, of radii r and R respectively, touch each other as shown in figure. The line l, which is parallel to the line joining the centres of

C_{1}

C_{2}

, is tangent to

C_{1}

at P and intersects

C_{2}

R^{2} = 2 r^{2}

∠ A O B

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