Question

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 

Solution

Correct option is

a2 > 2b2.

Equation of the given circle is 

                  2x (x – a) + y (2y – b) = 0   

          or      2x2 – 2ax + 2y2 – by = 0  

          or        x2 + y2 – ax – b/2 y = 0                       …(1) 

Since the two chords are bisected by x-axis so let (h, 0) be the mid point where h has two real values. 

Equation of the chord is

                   T = S1  

                 

It passes through (ab/2)  

                

         

   

Since the value of h are real and distinct so

               B2 – 4AC > 0  

                     

                            

                                    

                          a2 – 2b2 > 0  

                          a2 > 2b2.

SIMILAR QUESTIONS

Q1

If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 – 3ax + dy – 1= 0 intersect in two distinct points P and Q then the line

5x + by – a = 0 passes through P and Q for:  

Q2

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:

Q3

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 

Q4

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 

Q5

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

Q6

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

Q7

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

Q8

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

Q9

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

Q10

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: