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.

Why Kaysons ?

Video lectures

Access over 500+ hours of video lectures 24*7, covering complete syllabus for JEE preparation.

Online Support

Practice over 30000+ questions starting from basic level to JEE advance level.

Live Doubt Clearing Session

Ask your doubts live everyday Join our live doubt clearing session conducted by our experts.

National Mock Tests

Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation.

Organized Learning

Proper planning to complete syllabus is the key to get a decent rank in JEE.

Test Series/Daily assignments

Give tests to analyze your progress and evaluate where you stand in terms of your JEE preparation.



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.


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  






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 


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


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


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


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


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


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 


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:


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.


The normal 3x – 4y = 4 and 6x – 8y – 7 = 0 are tangents to the circle. Then its radius is: