Question

 be a given circle. Find the locus of the foot of perpendicular drawn from origin upon any chord of Swhich subtends a right angle at the origin.

Solution

Correct option is

 

Given circle is S ≡ x2 + y2 + 2gx + 2fy + c = 0. Let M (hk) be the foot of perpendicular from O (0, 0) to the chord AB.

  

Equation of AB, which passes through M (hk), is

              

  

or        hx + ky = h2 + k2  

Since AB subtends an angle 90o at origin. The combined equation of OAand OB is obtained by making the equation of circle homogeneous with the help of (1),

Since  ∠AOB = 90o  

  

  

       

SIMILAR QUESTIONS

Q1

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.

Q2

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.

Q3

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.

Q4

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.

Q5

 

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. 

Q6

 

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).   

Q7

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

Q8

 

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).

Q9

The centre of the circle S = 0 lie on the line 2x – 2y + 9 = 0 and S = 0 cuts orthogonally the circle x2 + y2 = 4. Show that circle S = 0 passes through two fixed points and find their co-ordinates.

Q10

 be a given circle. Find the locus of the foot of perpendicular drawn from origin upon any chord of Swhich subtends a right angle at the origin.