Tangents are drawn to the circle from a point which always lies on the line .Prove that the locus of the mid-point of the chords of contact is .
Given circle is
Let this be 1.
Given line is
Let this equation be 2.
Let (h,k) be a point on the line 2, hence .
This is equation 3
The equation of the chord of contact of tangents from (h,k)to the circle is
Let this be equation 4
Let (x1,y1) be the mid point of the chord, then teh equation of the chord is
Le this be 5.
The equations 4 and 5 represent the same.
Substituting these values in the equation 3, we get
Hence the locus of (x1,y1) is