Question

The equation of the locus of the mid points of the chords of the circle 4x2 + 4y2 – 12x + 4y + 1 = 0 that subtend an angle of  at its centre is:

Solution

Correct option is

Equation of the given circle is

                   4x2 + 4y2 – 12x + 4y + 1 = 0

 or                 x2 + y2 – 3x + y +  = 0                         …(1)   

Comparing (1) with

                   x2 + y2 + 2gx + 2fy + c = 0  

              

            

              

Centre of the circle is    

Radius of the circle is    

                                        

If m (hk) is the middle point of the chord AB which subtend 120o at the centre C then

                                                                                 

                           

                        

                 

  

                       

                       

                      

                       

                      

So the locus of the point (hk) is  

                    

SIMILAR QUESTIONS

Q1

The intercept on the line y = x by the circle x2 + y2 – 2x = 0 is AB. Equation of the circle on AB as diameter is:

Q2

A square is formed by following two pairs of straight lines y2 – 14y + 45 = 0 and x2 – 8x + 12 = 0. A circle is inscribed in it. The centre of the circle is 

Q3

Let PQ and RS be tangents at the extremities the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals 

Q4

A diameter of x2 + y2 – 2x – 6y + 6 = 0 is a chord to circle (2, 1), then radius of the circle is 

Q5

If α, β, γ, δ be four angles of a cyclic quadrilateral taken in clockwise direction then the value of  will be:

Q6

A square is inscribed in the circle x2 + y2 – 2x + 4y + 3 = 0. Its sides are parallel to the co-ordinate axes. Then one vertex of the square is

Q7

A circle passes through the point (–1, 7) and touches the line y = x at (1, 1). Its diameter is

Q8

Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is

Q9

Let a circle be given by 2x(x – a) + y(2y – b) = 0, (a  0, b ≠ 0). Find the condition on a and b, if two chords each bisected by the x – axis can be drawn to the circle from 

Q10

Find the radius of the smallest circle which touches the straight line 3x– y = 6 at (1, –3) and also touches the line y = x. complete up to one place of decimal.