JEE Advance - Mathematics (1983 - No. 51)
Through a fixed point (h, k) secants are drawn to the circle $$\,{x^2}\, + \,{y^2} = \,{r^2}$$. Show that the locus of the mid-points of the secants intercepted by the circle is $$\,{x^2}\, + \,{y^2} $$ = $$hx + ky$$.
The locus of the mid-points of the secants is a straight line.
The equation of the locus is obtained by substituting the coordinates of the midpoint into the equation of the circle.
The locus is a circle passing through the origin and (h,k).
The given equation $x^2 + y^2 = hx + ky$ represents a circle with center at (h/2, k/2) and radius √(h^2 + k^2)/2.
The locus represents a parabola.
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