. Plane and solid analytic geometry; an elementary textbook. It 27. Find the locus of the middle points of chords whichpass through a fixed point Qxv y^) of the circle x2 + y2 = r2. Let P, (V, ?/), be the middle point of any chord throughPv (xv y^). Let (xv y2~) be the coordinates of P2, theother extremity of the chord. From the formulas for bisecting a line [4], we have. (1) *: X* -\~ Xn and (2) y = U±±]h. And, since P2 is a point onthe circle, (3) x* + y*=r*.Here are three equationsbetween the variables x and y\ the constants xv yv and r,and the parameters and y2. It is therefore possibl
. Plane and solid analytic geometry; an elementary textbook. It 27. Find the locus of the middle points of chords whichpass through a fixed point Qxv y^) of the circle x2 + y2 = r2. Let P, (V, ?/), be the middle point of any chord throughPv (xv y^). Let (xv y2~) be the coordinates of P2, theother extremity of the chord. From the formulas for bisecting a line [4], we have. (1) *: X* -\~ Xn and (2) y = U±±]h. And, since P2 is a point onthe circle, (3) x* + y*=r*.Here are three equationsbetween the variables x and y\ the constants xv yv and r,and the parameters and y2. It is therefore possible 96 ANALYTIC GEOMETRY [Ch. VIII, § 61 to eliminate the parameters and obtain a single equationin terms of the variables and constants only. Solving (1)and (2) for x2 and yv we have x2 = 2 x - xv and y2 = 2 y - yv Substituting these values in (3), we have 4 x2 + 4 y2 — 4 xxx — 4 yxy + xx2 + yx2 = r2. But xx2 -j- yx2 = r2, and, dropping primes, the equation reduces to x2 + y2-x1x-y1y = 0. ? This is the equation of the locus of P. It is a circle on OPi as a diameter, since its centre is at the point [ -i, ^and it passes through the origin. When, as in the above problem, we have to determinethe locus of a point situated on a moving line whichrevolves about some fixed point in it, polar coordinatesare often convenient. The fixed point is taken as thepole, and
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