Question

Let 1 and C2 be two circles with C2 lying inside C1. A circle C lying inside C1 touches C1 internally and C2 externally. Identify the locus of the centre of C.

Solution

Correct option is

(ab) and (0, 0)

 

Let the given circles C1 and C2 have centres O1 and O2 with radii r1 andr2 respectively. Let centre of circle C is at O radius is r.

            

Which is greater than O1O2 as O1O2 < r1 + r2.

 

Alternative Method:

               

                 

Adding (1) and (2) we get

                        

Which represents an ellipse with foci are at (ab) and (0, 0).

SIMILAR QUESTIONS

Q1

P is a variable on the line y = 4. Tangents are drawn to the circle x2 + y2= 4 from P to touch it at A and B. The perpendicular PAQB is completed. Find the equation of the locus of Q.

Q2

 

Find the condition on abc such that two chords of the circle

                x2 + y2 – 2ax – 2by + a2 + b2 – c2 = 0  

passing through the point (ab + c) are bisected by the line y = x.  

Q3

 

Find the limiting points of the circles 

       (x2 + y2 + 2gx + c) + λ(x2 + y2 + 2fy + d) = 0

Q4

The circle x2 + y2 – 4x – 8y + 16 = 0 rolls up the tangent to it at  by 2 units, assuming the x-axis as horizontal, find the equation of the circle in the new position.  

Q5

 

Find the equation of the circle of minimum radius which contains the three circles 

                   x2 – y2 – 4y – 5 = 0 

               x2 + y2 + 12x + 4y + 31 = 0  

and         x2 + y2 + 6x + 12y + 36 = 0

Q6

Find the equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius.

Q7

 

If the line x cos α + y sin α = p cuts the circle x2 + y2 = a2 in M and N, then show that the circle, whose diameter is MN, is 

              

Q8

A line meets the co-ordinate axes at A and B. A circle is circumscribed about the triangle OAB. If the distance of the points A and B from the tangent at O, the origin, to the circle are m and n respectively, find the equation of the circle.

Q9

Tangents PQPR are drawn to the circle x2 + y2 = 36 from the point P(–8, 2) touching the circle at QR respectively. Find the equation of the circumcircle of the âˆ†PQR.

Q10

The circle x2 + y2 – 4x – 4y + 4 = 0 is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is x + y – xy + k(x2 + y2)1/2 = 0. Find k.