Question

Let S  x2 + y2 + 2gx + 2fy + c = 0 be a given circle. find the locus of the foot of the perpendicular drawn from the origin upon any chord of S which subtends a right angle at the origin.

Solution

Correct option is

Equation of the given circle is 

                   S = x2 + y2 + 2gx + 2fy + c = 0              …(1)

Any chord of the circle be 

                   lx + my = 1                                            …(2)  

Chord (2) subtends a right angle at the origin  So making equation (1) homogeneous with the help of (2), we get  

          (x2 + y2) + (2gx + 2fy) (lx + my) + c(lx + my)2 = 0  

          x2 + y2 + 2glx2 + 2gmxy + 2flxy + 2fmy2 + cl2x2 + cm2y2 + 2clmxy = 0  

             

Apply the condition of perpendicularity 

We get

                   

  

Any line through origin ⊥ to (1) is  

                                                                 mx – ly = 0            …(4)  

Both (2) and (4) give the foot of perpendicular whose locus is obtained by eliminating the variable lm between (2), (3) and (4).

On solving (2) & (4)

We get,      

                         

Putting in (2) we get

             

                

                         

  

Putting the value of l & m in  (3) we get  

             

  

 

SIMILAR QUESTIONS

Q1

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Q2

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Q3

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Q4

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Q5

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

Q6

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

Q7

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 

Q8

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:

Q9

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.

Q10

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