Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

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ANALYTIC GEOMETRY

2. A line-segment AB of fixed length moves so that its
extremities lie always on two perpendicular lines. Find the
locus of the point dividing AB in the ratio 2:1, using as
auxiliary variable the angle which the moving line makes with
one of the two perpendicular lines.
Ans. An ellipse, center in the point of intersection of the
given lines, axes along them, with length and breadth in the
ratio 2: 1.
3. Determine the locus described in Ex. 2, when the given
ratio is m1: m2.
4. The line L is the perpendicular bisector of the fixed
horizontal line-segment AB. The points R and S are taken
on L, with R always below S, so that the distance RS is one
half the distance AB. Find the locus of the point of intersection of AR and BS, taking the axes of x and y along AB
and L. Ans. The hyperbola, 2 xy + x2 = a2, through A and B.
5. A variable line is drawn through a fixed point P1 meeting a fixed line L in P2. Points P are taken on this line so
that the product of the distances P1P and P1P2 is constant.
Find the locus of these points.
Ans. Two circles, tangent at P1 to the line through P1 parallel to L.
6. Find the locus of points from which the tangents drawn
to a parabola are perpendicular.
Suggestion. Use the equation of the tangent with given
slope, Ch. IX, ~ 6, eq. (10).
7. The same for the ellipse.
8. The same for the hyperbola.
9. Determine the locus of the mid-points of all the chords
drawn from the vertex of a parabola.
3. Coordinates of a Point Tracing a Curve, as Auxiliary Variables. In the problems of the previous paragraph one of the
conditions governing the motion of the point tracing the locus
was the auxiliary motion of a line or of a second point. In