Curves in the Plane - Analytic Geometry and Vectors - University Mathematics Handbook

University Mathematics Handbook (2015)

III. Analytic Geometry and Vectors

Chapter 2. Curves in the Plane

The equation in two variables , , is called a curve equation if, and only if, all points of the curve hold it. That is, points not on the curve do not hold this equation.

A major purpose of analytical geometry is analyzing curves through their equations.

2.1  Straight Line

First-order equation is a straight line equation.

If is the angle between a straight line and the positive part of -axis, measured from -axis to the straight line counterclockwise, then the tangent of the angle is called the slope of the line.

The slope of a straight line is , .

If , the line has no slope.

a.   is the equation of a line with a slope .

b.   is a line parallel to the -axis (its slope is )

c.   is a line parallel to -axis (it has no slope).

d.   is a straight line passing through the origin.

e.  The equation of a line passing through points and is

.

f.  The distance of point from the straight line is

.

g.  Let there be two straight lines, , and .

1.  If , then they are parallel to each other.

2.  If , then they are perpendicular to each other.

3.  , where is the angle between the lines.

2.2  Circles

a.  A circle is the locus of points for which the distance from a given point is constant, called a radius.

b.  The equation of circle for which the center is in point and its radius is is

(*)

c.   is an equation of a circle if . In such a case, its center is point , and its radius is

d.  The equation of a tangent line to circle (*) at the point is

e.  Straight line is tangent to circle (*) if, and only if

2.3  Canonical Ellipses

a.  An ellipse is the locus of points M for which the sum of distance to two points and , called foci, is constant .

b.  An ellipse is canonical if its foci are on -axis and symmetric to each other about -axis.

c.  Canonical ellipse equation: , where , are its foci, , and are its semi-axes, and .

d.  The equation of a tangent line to ellipse at the point is .

e.  Its eccentricity is .

f.  , are focal radii for point , such that .

g.  The tangent to ellipse forms equal angles with the focal radii of point of tangency, such that .

h.  The straight line is tangent to canonical ellipse if, and only if, .

i.   is the equation of ellipse in polar coordinate system.

2.4  Hyperbolas

a.  A hyperbola is a locus of points for each of which the absolute value of the difference between the distances to two given points and , called foci, is constant .

b.  A hyperbola is canonical if its foci are on the -axis and symmetric to each other about -axis.

c.  A canonical hyperbola equaton is , where the foci are , (Figure 1).

is its real semi-axis, while is its imaginary semi-axis, and .

d.  The lines and are called the asymptotes of the hyperbola.

Figure 1

e.   is the conjugate hyperbola of a canonical hyperbola (Figure 2).

Figure 2

f.  Its eccentricity is .

g.  , are the focal radii of point , such that .

h.  The tangent to a hyperbola at point of tangency bisects the angle .

i.  The equation of tangent line to canonical hyperbola at the point is .

j.  Line is tangent to canonical hyperbola if, and only if, .

k.   is the canonical parabola equation relative to a polar coordinate system.

2.5  Parabolas

a.  A parabola is the locus of points for which the distance to a given point , called parabolic focus is equal to their distance to a given straight line called directrix, such that .

b.  A parabola is called canonical if its focus is on the -axis, its directrix is perpendicular to the -axis, and the focus and perpendicular are on both sides of the origin, at equal distances from it.

c.  The canonical parabola equation is , where the focus is at and the equation of its directrix is .

d.  The equation of a tangent line to a parabola at point of tangency is .

e.  Straight line is a tangent to canonical parabola if, and only if, .

f.   is the focal radius of , .

g.   if is tangent to the parabola.

h.   is the equation of parabola relative to polar coordinates.

2.6  Conic Sections

Circle, ellipse, hyperbola, parabola, and two intersecting straight lines are obtained by intersection of a plane with a right circular cone.

Theorem: Let be the angle between the generator of the cone and its axis. Then:

a.  If a plane does not pass through the vertex of a cone, forming angle with the cone axis, then the line of intersection is

1. A circle, when

2. An ellipse, when (Figure 1).

3. A parabola, when (Figure 2).

4. A hyperbola, when (Figure 3).

5. Two intersecting straight lines, when (Figure 4).

   Figure 1    Figure 2    Figure 3


b.  If the plane passes through the vertex, a point, or a straight line, a pair of straight lines may be obtained by the intersection (Figure 4).

A point, or a degenerate circle

A straight line, or a degenerate parabola

Intersection of two lines, or a degenerate hyperbola

Figure 4