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Grade: 11
        
the feet of perpendicular from the origin on a variable chord of the circle X2 + Y2 -2 X - 2y = 0 is  N.  if the variable chord makes an angle of 90 degree at the origin then the locus of N  has the equation....?
6 months ago

Answers : (2)

Dr Bhishma Hazarika
23 Points
							
Let (a,b) is the foot of the perpendicular drawn on the variable chord of the circle from the origin.
 
 So, Equation of the chord  y-b=-a/b(x-a)
                                        or ax+by={a}^2 +b^2
                                        or  \frac{ax+by}{a^2+b^2 }=1….....................(1)
 
 Now hoogeneous form of the given circle by equation is     x^2+y^2-2x\frac{ax+by}{a^2+b^2}-2y\frac{ax+by}{a^2+b^2}=0  
 
 
It will represent pair of st.lines. but by question chord makes 900 at the origin;
           
    so  co-officiant of x^2+co-officiant of y^2 is zero
 
    i.e    
 
   
5 months ago
Dr Bhishma Hazarika
23 Points
							
Continue from the first answer......
 
 
(1-\frac{2a}{a^2+b^2})+(1-\frac{2b}{a^2+b^2})= 2(a^2+b^2)-2a-2b=0
 
 
so locus of the foot of the perpendicular from the orogin is another circle
 
  2(x^2+y^2)-2x-2y=0
 
 
5 months ago
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