Question

Find the locus of the mid point of the chord of the circle x2 + y2 = a2which subtend a right angle at the point (pq).

Solution

Correct option is

 

Let the mid point of the chord be (hk) and circle be represented by 

                   

Then equation of chord is T = S1

                hx + ky – a2 = h2 + k2 – a2

or            hx + ky = h2 + k2                       …(1)

 

OM = length of perpendicular from O (0, 0) to the line (1)

                                  

    

Since M is the mid point of AB

∴ MA = MB = MP (since distances from the mid point of hypotenuse of right angled triangle of the vertices are equal)

   

⇒             a2 – h2 – k2 = (h – p)2 + (k – q)2  

⇒              2h2 + 2k2 – 2ph – 2qk + p2 + q2 – a2 = 0

Hence locus of mid point M (hk) is

                 2x2 + 2y2 – 2px – 2qy + p2 + q2 – a2 = 0  

SIMILAR QUESTIONS

Q1

 

Find the equation of a circle which passes through the point

(2, 0) and whose centre is the limit of the point of intersection of the lines 3x + 5y = 1and (2 + c)x + 5c2y = 1as c → 1.

Q2

Tangents are drawn from (6, 8) to the circle x2 + y2 = r2. Find the radius of the circle such that the areas of the âˆ† formed by tangents and chord of contact is maximum.

Q3

Find the radius of smaller circle which touches the straight line 3x – y = 6 at (1, –3) and also touches the line y = x.

Q4

2x – y + 4 = 0 is a diameter of the circle which circumscribed a rectangleABCD. If the co-ordinates of A and B are A(4, 6) and B(1, 9), find the area of rectangle ABCD.

Q5

Let 2x2 + y2 – 3xy = 0 be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

Q6

If the circle C1x2 + y2 = 16 intersects another circle C2 of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to (3/4), find the co-ordinates of centre C2.

Q7

The circle x2 + y2 = 1 cuts the x-axis at P and Q. another circle with centre at Q and variable radius intersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum area of the triangleQSR.

Q8

 

Find the equation of a circle having the lines x2 + 2xy + 3x + 6y = 0 as its normals and having size just sufficient to contain the circle

                         x(x – 4) + y(y – 3) = 0. 

Q9

 

Find the equation of the circle whose radius is 5 and which touches the circle 

              x2 + y2 – 2x – 4y – 20 = 0 at the point (5, 5).   

Q10

 

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 (ab/2).