Question

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.

Solution

Correct option is

1

 

The given circle is x2 + y2 – 4x – 4y + 4 = 0. This can be re-written as (x– 2)2 + (y – 2)2 = 4 which has centre C (2, 2) and radius 2.

Let the equation of third side is  

Length of perpendicular from (2, 2) on AB = radius = CM

∴                             

Since origin and (2, 2) lie on the same side of AB

  

  

Hence AB is the diameter of the circle passing through ∆OAB, mid point of AB is the centre of the circle i.e., 

Substituting the values of a and b in (1) then

                         

∴ Locus of (hk) is   

                        x + y – xy +   

Hence the required value of k is 1.

SIMILAR QUESTIONS

Q1

 

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.  

Q2

 

Find the limiting points of the circles 

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

Q3

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.  

Q4

 

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

Q5

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

Q6

 

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 

              

Q7

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.

Q8

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.

Q9

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.

Q10

Tangents TP and TQ are drawn from a point T to the circle x2 + y2 = a2. If the point T lies on the line px + qy = r, find the locus of centre of the circumcircle of triangle TPQ.