Question

A circle C1 of radius b touches the circle x2 + y2 = a2 externally and has its centre on the positive x-axis; another circle C2 of radius c touches the circle C1 externally and has its centre on the positive x-axis. Given ab < c, then the three circles have a common tangent if abc are in 

Solution

Correct option is

G.P.

The centre of C1 is (a + b, 0) and the centre of C2 is

(a + 2b + c, 0)

 
 

 

 

 

 

 

 

 

 

Let y = mx + k be a tangent common to the three circles.

Since it touches x2 + y2 = a2C1 and C2

                             

As the centre of the three circles lie on the same side of the line y = mxk, taking the same sign, say positive, in the three relations we get,     

  

   

  

SIMILAR QUESTIONS

Q1

A triangle PQR is inscribed in the circle x2 + y2 = 25. If Q and R have coordinates (3, 4) and (–4, 3) respectively, then QPR is equal to

Q2

For each natural number k, let Ck denote the circle with radius centimeters and centre at the origin O, on the circle Ck a particle moves k centimeters in the counter-clockwise direction. After completing its motion on Ck, the particle moves to Ck + 1 in the radial direction. The motion of the particle continues in this manner. The particle starts at (1, 0). If the particle crosses the positive direction of x-axis for the first time on the circle Cn then n =

Q3

If the area of the quadrilateral formed by the tangent from the origin to the circle x2 + y2 + 6x – 10y + c = 0 and the pair of radii at the points of contact of these tangents to the circle is 8 square units, then c is a root of the equation

Q4

If two distinct chords, drawn from the point (pq) on the circle x2 + y2px + qy (where pq  0) are bisected by the x-axis, then

Q5

Let A0 A1 A3 A4 A5 be a regular hexagon inscribed in a unit circle with centre at the origin. Then the product of the lengths of the line segmentsA0 A1A0 A2 and A0 A4 is

Q6

C1 and C2 are circles of unit radius with centres at (0, 0) and (1, 0) respectively. C3 is a circle of unit radius, passes through the centres of the circles C1 and C2 and have its centre above x-axis. Equation of the common tangent to C1 and C3 which does not pass through C2 is

Q7

A chord of the circle x2 + y2 – 4x – 6y = 0 passing through the origin subtends an angle tan-1 (7/4) at the point where the circle meets positivey-axis.

Equation of the chord is

Q8

On the line joining the points A (0, 4) and B (3, 0), a square ABCD is constructed on the side of the line away from the origin. Equation of the circle having centre at C and touching the axis of x is 

Q9

A circle with centre at the origin and radius equal to a meets the axis of x at A and B.P (α) and Q (β) are two points on this circle so that α – β = 2γ, where γ is a constant. The locus of the point of intersection of APand BQ is

Q10

Angle of intersection of these circle is