Question

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

Solution

Correct option is

p2 > 8q2

Let PQ be a chord of the given circle passing through P(pq) and the coordinates of Q be (xy). Since PQ bisected by the x-axis, the mid-point of PQ lies on the x-axis which gives y = –q.  

Now Q lies on the circle x2 + y2 – px – qy = 0  

So            x2 + q2 – px + q2 = 0  

                                                                        

Which gives two values of x and hence the coordinates of two points and R (say), so that the chords PQ and PR are bisected by x-axis. If the chords PQ and PR distinct, the roots of (i) are real distinct.

SIMILAR QUESTIONS

Q1

An equation of the chord of the circle x2 + y2 = a2 passing through the point (2, 3) farthest from the centre is

Q2

The lengths of the intercepts made by any circle on the coordinates axes are equal if the centre lies on the line (s) represented by

Q3

A circle touches both the coordinates axes and the line  the coordinates of the centre of the circle can be

Q4

If the tangent at the point P on the circle x2 + y2 + 6x + 6y = 2 meets the straight line 5x – 2y + 6 = 0 at a point Q on the y-axis, then the length ofPQ is

Q5

If a > 2b > 0 then the positive value of m for which  is a common tangent to x2 + y2 = b2 and (x – a)2y2 = b2 is   

 

Q6

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

Q7

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

Q8

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 =

Q9

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

Q10

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