0-15 TRIGONOMETRY 31
PROBLEMS
1. Grap, hand find the focus, vertex, and directrix:
(a) y = x2 2x + 5 b) y -X2 - 4x + 7
(c) - 3 = 2(x — 1)2 (d) 2 - 2x + 3 = 0
(e) y" - 2 - 3 = O (f) x2 = 0
2. Find the center and radius of each of the following circles and graph:
(a) x2 + Y2 = 7 i 2x 22y2 5
(c) x +2—5x 6y=O (d) 3x2+3y2+4x + 6y+52=O
3. Show that a circle x" + y2 + ax + by + c - 0 is tangent to the x-axis if, and only
if, 4c = a2.
4. Find the equation of a straight line of slope 2 tangent to the circle X2 + y2 = 180.
5. Find the equation of the circle through the points (1, 1), (1, -2), and (2, 3).
6. Find the points of intersection of the circles x2 + y - 2x - 0, x2 + y - 3y = 0.
7. Let to circles x2 y ax +0 b+ / +, x 4 2 Ax + By + C = O be
given and form the equation:
A x2+ 2+ax+by - c-(x2`+y2'+Ax+ By+QC)=0
(a) Show1 that if the circles meet in a single point, then (*) represents a common
tatgent.
(b) Show that, if the circls do not intersect and are not concentric, then (*) represents a line perpendicular to the line that joins the centers of the circles.
8. F1ind the vertexes, center, eccentricity, faci, and directrices, and graph:
(a) 3x` + 4y2 = 12 (b) 9x2 + 5y2 = 45
c) 5x2 - 4Y" = 20 (d) 92 -l16y2 =12
9. Find the equation of an ellipse that satisfies the stated conditions and graph:
a) Focal tdistanc 4, major axis 6.
(hb Major axis 4. minor axis 2
(c) Ecen tricity maj, ujor axis 4.
10. Reduce to a standard form for an ellipse or hyperbola and graph:
(a) 2x24y" -3x-4y—2=0 (b) 3x-62+3x+- y-2= 0
(c) 4x2Yx" I 6x + 2y + 15=0 (d) x2 2y2 - 2x + 12y + 19 =0
11, Let a point (x, y) move so that the ratio of its distance from (c, 0) to the line x = a e
is e, where a, c, e are given positive numbers, e 1, and c = ae. Show that the locus
is an ellipse or hyperbola with center at (0, 0) according as e < 1 or c > 1. Show
further that every ellipse and hyperbola with center (0, 0) and foci on the x-axis is
obtained in this way. Show that the same curve is obtained if (c, 0) is replaced by
( - c, 0 and the line x = a te by the line x = - ae.
12. Show that a straight line in the xy-plane meets a nondegenerate conic section in at
most two points. Describe conditions under which it can meet the conic section in
just one point.
0-15 TRIGONOMETRY
Angles can be measured in degrees or in radians. One radian is the angle
subtended at the center of a circle by an are equal to the radius; hence, in a
circle of radius 1, the are has length 1. For a central angle of a radians the are
is ra (see Figure 0-24). An angle of d degrees has radian measure a, where
