4.1 Geometry 219618.Find the locus of the projection of a fixed point on a circle onto the tangents to the
circle.
619.On a circle of centerOconsider a fixed pointAand a variable pointM. The circle
of centerAand radiusAMintersects the lineOMatL. Find the locus ofLasM
varies on the circle.
620.The endpoints of a variable segmentABlie on two perpendicular lines that intersect
atO. Find the locus of the projection ofOontoAB, provided that the segmentAB
maintains a constant length.
621.From the center of a rectangular hyperbola a perpendicular is dropped to a variable
tangent. Find the locus in polar coordinates of the foot of the perpendicular. (A
hyperbola is called rectangular if its asymptotes are perpendicular.)
622.Find a transformation of the plane that maps the unit circlex^2 +y^2 =1 into a
cardioid. (Recall that the general equation of a cardioid isr= 2 a( 1 +cosθ).)
623.Prove that the locus described by the equationx^3 + 3 xy+y^3 =1 contains precisely
three noncollinear pointsA, B, C, equidistant to one another, and find the area of
triangleABC.
624.Fornandptwo positive integers consider the curve described by the parametric
equations
x=a 1 tn+b 1 tp+c 1 ,
y=a 2 tn+b 2 tp+c 2 ,
z=a 3 tn+b 3 tp+c 3 ,wheretis a parameter. Prove that the curve is planar.625.What is the equation that describes the shape of a hanging flexible chain with ends
supported at the same height and acted on by its own weight?
4.1.4 Coordinate Geometry in Three and More Dimensions..........
In this section we emphasize quadrics. A quadric is a surface in space determined by a
quadratic equation. The degenerate quadrics—linear varieties, cones, or cylinders over
conics—add little to the picture from their two-dimensional counterparts, so we skip them.
The nondegenerate quadrics are classified, up to an affine change of coordinates, as
- x^2 +y^2 +z^2 =1, ellipsoid;
- x^2 +y^2 −z^2 =1, hyperboloid of one sheet;