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

A SECOND CHAPTER ON LOCI

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11. Find the locus of the point of intersection of the line
drawn through the vertex of the parabola y2 = 2 mx perpendicular to a variable tangent and the line drawn through the
point of tangency perpendicular to the axis.
Ans. The semi-cubical parabola, my2= 2 x3.
12. A vertex 0 of a quadrilateral and the directions of the
sides through 0 are fixed. The two angles adjacent to 0 are
right angles and the diagonal joining their vertices has a fixed
direction. Find the locus of the fourth vertex.
Ans. Straight line through 0, perpendicular to the line
through 0 which makes an angle with the fixed direction equal
to the sum of the two angles which the sides through 0 make
with the fixed direction.
13. A parallelogram has sides of constant length a and b
and has one vertex fixed at a point 0. It opens and closes
so that the two sides through 0 are always equally inclined
to a fixed line through 0. Taking the angle which these sides
make with the fixed line as auxiliary variable, find the locus
of the vertex opposite to 0.
14. Each of two straight lines moves always parallel to
itself so that the product of the distances of the lines from a
fixed point 0 is constant. Find the locus of their point of
intersection, taking the axes so that 0 is the origin and the
directions of the two lines are equally inclined to the axis of x.
Ans. Two conjugate hyperbolas, center at 0, with asymptotes parallel to the fixed directions.
15. Find the locus of the center of a circle which passes
through a fixed point on one of two perpendicular lines and
intercepts a segment of constant length on the other.
16. Find the locus of points from which it is possible to
draw two perpendicular normals to a parabola.
17. Find the locus of the point of intersection of the tangents
-o an ellipse at points subtending a right angle at the center.
18. The preceding problem for the hyperbola.