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A variable chord of circle x^(2)+y^(2)+2...

A variable chord of circle `x^(2)+y^(2)+2gx+2fy+c=0` passes through the point `P(x_(1),y_(1))`. Find the locus of the midpoint of the chord.

Text Solution

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Let `M(h,k) ` be the midpoint of the variable chord AB.
`:. `Equation of the chord AB is
`hx+ky+g(x+h)+f(y+k)+c=h^(2)+k^(2)+2gh+2fk+c` (Using `T=S_(1))`
Since chord AB passes through `P(x_(1),y_(1))`, we have
`hx_(1)+ky_(1)+g(x_(1)+h)+f(y_(1)+k)+c=h^(2)+k^(2)+2gh+2fk+c`
or `hx_(1)+ky_(1)+gx_(1)+fy_(1)=h^(2)+k^(2)+gh+fk`
`:. `Equation of locus of point M is
`x x _(1)+y y_(1)+gx_(1)+fy_(1)=x^(2)+y^(2)+gx+fy`
Alternative method `:`
Since `CM _|_AB`, slope of CM ` xx` Slope of AB `= -1`
`:. (h=g)/(k+f)xx(h-x_(1))/(k-y_(1))=-1`
or `(x+g)(x-x_(1))+(y+f)(y-y_(1))=0`
or `x^(2)+y^(2)+gx+fy-gx_(1)-fy_(1)-x x_(1)-y y_(1)=0` , which is equation of required locus.
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