Analytic geometry. Textbook
.pdf4.1. Straight lines on the plane |
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9. Find equations of the lines through the following points and parallel |
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to the y-axis: |
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a) (2; 3); |
b) (7; 4); |
c) (0; p); |
d) ( p; q). |
10. Given that the line kx y = 4 is orthogonal to the line kx+9y = 11,nd the value of k.
11.Find the equation of the straight line passing through the point ( 2; 3) and parallel to the straight line 2x y 9 = 0.
12.Find the equation of the straight line passing through the point (3; 2) and parallel to the straight line passing through the points (4; 1) and (3; 2).
13.Find the equation of the straight line passing through the point (3; 1) and orthogonal to the straight line 2x y + 9 = 0.
14.Find the distance of the point:
a) A(4; 2) |
from the straight line |
8x 15y 11 = 0; |
b) B(2; 7) |
from the straight line |
12x + 5y 7 = 0; |
c) C( 3; 5) |
from the straight line |
9x 12y + 2 = 0; |
d) D( 3; 2) |
from the straight line |
4x 7y + 26 = 0; |
e) E(8; 5) |
from the straight line |
3x 4y 15 = 0. |
15. Find the distance from the origin to the straight line
a) 3x 4y 10 = 0, b) 12x + 5y 39 = 0.
16. Find the equation of the straight line with the slope 2 which is 3 units from the origin.
17.Find the equation of the straight line if it is orthogonal to the straight line 3x 4y + 10 = 0 and is 2 units from the point (2; 3).
18.Find the equation of the straight line passing through the point (2; 1) if this line is 2 units from the origin.
19.Find the equation of the straight line if it is passes through the point ( 1; 4) and is 6 units from the point (3; 2).
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Chapter 4. Problems |
20. Find the equation of the straight line which passes through the point (3; 1) and has the slope k = 2.
21.Find the angle of inclination of the straight line passing through the points (3; 1) and ( 2; 4).
22.Find the equation of the straight line passing through the point (3; 1) and having the slope 3.
23.Find the equation of the straight line passing through the point (5; 6) if the x-intercept of this line is 2.
24.Find the equation of the straight line passing through the point ( 3; 4), if
a)the x-intercept is 10;
b)the sum of intercepts is 12;
c)the product of intercepts is 50;
d)this line is 3 units from the origin;
e)this line is 5 units from the point (12; 9);
25.Find the equation of the straight line with the slope 4=3, if
a)the x-intercept is 6;
b)it is 6 units from the origin;
c)it is 4 units from the point (10; 2);
d)it is equidistant from the points (2; 6) and (3; 8).
26.Find the equation of the straight line
a)passing through the point (4; 2) and having equal intercepts;
b)which is perpendicular to the straight line 4x y 7 = 0 at the point of the given line whose abscissa is 1;
c)which has equal intercepts and is tangent to the circle with center (2; 1) and radius 5;
d)which is parallel to the straight line x + 2y 7 = 0 and tangent to the circle x2 + y2 = 5.
4.1. Straight lines on the plane |
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27.Find the equation of the straight line passing through the intersection point of the straight lines 3x 2y 13 = 0 and x + y 6 = 0 and passing through the point (2; 3).
28.Find the equation of the straight line passing through the intersection point of the lines 4x + y 7 = 0 and 3x 2y 10 = 0 and parallel to the line x 3y 6 = 0.
29.Find the equation of the straight line passing through the intersection point of the straight lines 3x + 5y 13 = 0 and x + y 1 = 0 and orthogonal to the straight line 7x 5y 10 = 0.
30.Determine the constants in the normal equation of the straight line
x cos + y sin p = 0, if the line passes through the point ( 4; 12), and
= 45 .
31.Determine the coe cients in the normal equation of the straight line x cos + y sin p = 0, if p = 10 and the line passes through the point (14; 2).
32. For each of straight lines passing through the given points nd
whether the angle of inclination is an acute, obtuse, or right angle.
a) (3; 5) and ( 4; 9); |
b) ( 2; 9) and (4; 3); |
c) (8; 3) and ( 2; 2); |
d) ( 4; 1) and (5; 8); |
e) (5; 2) and (5; 4); |
f) (3; 1) and (2; 2). |
33. Which of the following equations of lines have normal form?
a) 2x 3y + 4 = 0;
c) 23x 47y 1 = 0; e) 35x 45y 2 = 0;
g) x 4 = 0;
34. Reduce the equations of the straight lines to the normal form:
a) 4x 3y + 10 = 0; |
b) 5x + 12y 39 = 0; |
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c) 6x + 8y 15 = 0; |
d) x 2y + 3 = 0; |
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xp |
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= 4; |
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cos 10 + y |
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sin 10 + 4 = 0. |
e) y |
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f) x |
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114 Chapter 4. Problems
35. Reduce the equations of the straight lines to the normal form and
nd the corresponding angle and distance p of the origin from the line:
a) x + y + 2 = 0; |
b) x y + 4 = 0; |
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c) x + y 4 = 0; |
d) x y 6 = 0; |
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e) x + yp |
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= 6; |
f) x + yp |
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= 4; |
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3 |
3 |
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p |
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p |
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g) x 3 + y = 1; |
h) x 3 y = 2. |
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36. Find the equation of the straight line with the slope 2 and the x-intercept 3.
37. Determine the value of k, given that the straight line 3x ky 7 = 0 passes through the point (2; 1).
38.Determine the value of k, given that the slope of the straight line 3kx 2y 9 = 0 is 14.
39.Determine the value of k, given that the straight line 3x + ky = 3 has equal intercepts.
40.Given that the straight line Ax + By + 10 = 0 is parallel to the straight line 3x + y = 7 and meets the straight line x + y = 7 on the x-axis,
nd the values A and B.
4.2Second order curves on the plane
41.Find the center and radius of each of the circles represented by the
following equations, and draw each circle:
a) x2 + y2 2x 6y = 15, |
b) x2 |
+ y2 |
= 6x, |
c) x2 + y2 + 6x + 4y = 3, |
d) x2 + y2 2x + 8y = 8. |
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42.Find the points of intersection of the straight line 3x y = 3 with the circle x2 + y2 + x 4y 3 = 0, and draw these lines.
43.Find the intersection points of the straight line 2x + y + 3 = 0 with the circle x2 + y2 4x 6y = 7, draw these lines.
4.2. Second order curves on the plane |
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44. Find the intersection points of of the x-axis with the circle x2 + y2 2x + 4y = 8.
45. On the circle x2 + y2 6x + 2y = 7 nd each point whose ordinate is 3, and nd each point in which the circle cuts the x-axis, draw the circle and these points.
46. Find the equation of the circle with the center (2; 2) and radius 13. For each of the following points, establish whether it belongs to the circle,
lies outside the circle or inside the circle: |
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a) A(3; 10); |
b) B(14; 3); |
c) C(2; 11); |
d) D(12; 4); |
e) E(2; 15); |
f) F (12; 12). |
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47.Show that (8; 3) and (2; 5) are on a circle whose center is (5; 1) and nd the radius of the circle.
48.In each of these cases show that the point M is on the given circle,
and nd the equation of the tangent at this point: |
= 49, M(2; 5); |
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a) x2 + y2 = 25, M(3; 4); |
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b) x2 + y2 |
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c) x2 + y2 |
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d) x2 + y2 |
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21; 21p |
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= 34, M( 5; 3); |
= 1, M |
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e) x2 + y2 |
= 20, M( 4; |
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2); |
f) x2 + y2 = 37, M( |
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1; |
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6); |
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g)x2 + y2 6x + 2y = 0, M(2; 2);
h)x2 + y2 + 4x 7y 11 = 0, M(3; 2).
49.Find equations of tangents to the following circles under the stated conditions and nd the tangency points:
a)x2 + y2 = 25, the slope of the tangent is 34;
b)x2 + y2 = 49, the slope of the tangent is 125 ;
c)x2 + y2 = 36, the tangent is parallel to the straight line 4x = 3y;
d)x2 + y2 = 13, the tangent is orthogonal to the straight line x = 23y;
e)x2 + y2 = 104, the point ( 8; 12) is on the tangent;
f)x2 + y2 = 64, the tangent passes through the point (8; 4).
50.Prove that the circle x2 + y2 = 34 passes through the point (5; 3) and nd the equation of the tangent at this point.
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Chapter 4. Problems |
51. Prove that the circle x2 + y2 2x + 4y 5 = 0 passes through the point (2; 1) and nd the equation of the tangent at this point.
52.Find the equation of the circle with the center (5; 2) which is tangent to the x-axis, and draw this circle.
53.Find the equation of the circle with the center (4; 3) which is tangent to the y-axis, and draw this circle.
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54. Check that the lines 2x + 5 y 15 = 0 and 11 x 5y + 30 = 0 are the tangents of some circle with the center at the origin and nd the radius of this circle.
55. Find the radius r and the equation of the circle with the center
C(4; 3) and the tangent 3x 4y 20 = 0.
56. Find the radius and equation of the circle with the center C(3; 1) and the tangent y = 3x + 10.
57.Find the radius and equation of the circle with the center C(2; 4) and the tangent 3x 4y = 0.
58.Find the value k if the line y = kx + 10 is the tangent to the circle with the radius 6 and center (0; 0).
59.Find the equation of the straight line if it is parallel to the straight line 5x 12y 14 = 0 and is tangent to the circle x2 + y2 8x + 12y = 12.
60.Draw each of the following parabolas, in each case nd the coordinates of the focus and the equation of the directrix, mark the focus and draw the directrix a) y2 = 8x; b) y2 = 8x; c) x2 = 24y; d) x2 = 24y.
61.In the equation y2 = 2px nd the value of p under each of the following conditions:
a)the parabola passes through the point (2; 6);
b)the distance from the vertex to the focus is 4;
c)the focus is the point ( 5; 0).
4.2. Second order curves on the plane |
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62. Under each of the following conditions nd the equation of the parabola whose vertex is at the origin and whose axis is one of the coordinate axes:
a)the focus is (0; 3);
b)the focus is on the line 2x + 4y = 12;
c)the directrix is the line y = 6.
63.Find the coordinates of a point P on the parabola y2 = 4x if the distance of this point to the focus is 10.
64. Draw the graphs of each of the following pairs of equations and nd
the common points in each case: |
b) y2 = 3x, x 4y + 12 = 0; |
a) y2 = 9x, 3x 7y + 30 = 0; |
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c) x2 = 9y, 2x 3y + 3 = 0; |
d) y2 = 12x, x y = 9; |
e) x2 + y2 4x = 4, x = y2; |
f) x2 = 5y, y2 = 5x; |
g) x2 = 4, y2 = 9. |
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65.Find the points on the parabola y2 = 12x which are 6 units from the focus.
66.Find the points on the parabola y2 = 5x which are 6 units from the
vertex.
67.Find the equation of the parabolas with the following properties:
a)standard equation, the distance of the focus from the vertex is 3;
b)the point (5; 0) is the focus, the y-axis is the directrix;
c)the parabola is symmetric with respect to the x-axis and passes through the point (1; 4);
d)the parabola is symmetric with respect to the y-axis, the point (0; 2) is the focus, the origin is the vertex;
e)the parabola is symmetric with respect to the y-axis, passes through the origin and the point (6; 2).
68.On the parabola y2 = 8x nd a point whose focal distance is 20.
69.Find intersection points of the parabola y2 = 18x with the following straight lines:
118 |
Chapter 4. Problems |
a) 6x + y 6 = 0; |
b) 9x 2y + 2 = 0; |
c) 4x y + 5 = 0; |
d) y 3 = 0. |
70. Determine coordinates of the vertex of the parabola, the value of the parameter and the direction of the axis, if the parabola is de ned by
one of the following equations:
a) y2 10x 2y 19 = 0; b) y2 6x + 14y + 49 = 0;
p c) y2 + 8x 16 = 0; |
d) x2 6x 4y + 29 = 0; |
e) y = x2 8x + 15; |
f) y = x2 + 6x. |
71.Find equations of tangents to the following parabolas and the tangency points under the given conditions:
a)y2 = 16x, the tangent has the slope 2;
b)y2 = 12x, the tangent has the slope 13;
c)y2 = x, the tangent is parallel to the straight line x + 6y = 5;
d)3y2 = x, the tangent is orthogonal to the straight line x + y = 5;
e)y2 = 12x, the tangent passes through the point ( 2; 5) 1);
f)x = 2y2, the tangent passes through the point (8; 2);
g)y2 = 4x, the tangent has equal intercepts.
72.If the line x 2y +12 = 0 is tangent to the parabola y2 = cx, nd c.
73.Find equations of the tangents to the parabola y2 = 8x passing through the point (5; 7).
74.Check that the straight line x + 3y + 9 = 0 is the tangent to the parabola y2 = 4x and nd the tangency point.
75.Find equations of the tangents to the parabola y2 = 12x under the following conditions and nd the tangency points:
a)at the points of the parabola with the abscissa x = 3;
b)parallel to the straight line 3x y + 5 = 0;
c)orthogonal to the straight line 4x 2y + 9 = 0.
1)In this and the next cases the given point is not on the parabola.
4.2. Second order curves on the plane |
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76. Find the shortest distance from the parabola y2 = 64x to the
straight line 4x + 3y + 46 = 0.
77. The straight line x 2y + 5 = 0 is the tangent to the parabola y2 = 2px. Find the parameter p of the parabola and the tangency point.
78. For the ellipses de ned by the following equations, write the equation in the standard form, nd the focuses, eccentricity and equations of
their directrices: |
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a) 4x2 + 25y2 = 100; |
b) 9x2 |
+ 16y2 = 144; |
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c) 3x2 + 4y2 = 12; |
d) 3x2 |
+ 4y2 |
= 24; |
e) 25x2 + 9y2 = 225; |
f) 9x2 + 4y2 = 36; |
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g) x2 + 25y2 = 25; |
h) 6x2 |
+ 9y2 |
= 36; |
79. For the ellipse de ned by the equation 25x2 + 169y2 = 4225, nd the semiaxes, coordinates of the focuses and eccentricity.
80. Find the standard equation of the ellipse, if the distances from its focuses to one of the vertices lying on the major axis are 7 and 1.
81. Find equations of the directrices of the ellipse x2 + y2 = 1.
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82. Find the standard equation of the ellipse whose directrices are x = 8 and minor semiaxis is 4.
83. On the ellipse x2 + y2 = 1 nd a point that is 5 units from its small
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axis.
84. On the ellipse |
x2 |
+ y2 = 1 nd a point whose distance from the |
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100 |
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right focus is four times the distance from its left focus. |
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85. Find the intersection points of the ellipse x2 |
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y2 |
= 1 and the |
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36 |
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straight line 2x y 9 = 0.
86. Find the standard equations of the ellipses with the following properties, the vertices given below are on the major axis:
120 |
Chapter 4. Problems |
a)focuses ( 4; 0), vertices ( 6; 0);
b)focuses ( 3; 0), directrices x = 12;
c)minor axis 6, focuses ( 4; 0);
d)vertices ( 8; 0), eccentricity 34;
e)vertices (0; 8), eccentricity 34;
f)eccentricity 12, major axis 12, focuses on OY .
87.Find the standard equations of the ellipses with the following prop-
erties:
a) the major semiaxis is 4, the minor semiaxis is 2;
b) the distance between the focuses is 6, the major semiaxis is 5;
c) the major semiaxis is 10, eccentricity e = 0:8; p
d) the minor semiaxis is 3, eccentricity e = 2 =2;
e)the sum of semiaxes and the distance between the focuses equal 8.
88.Prove that the ellipses 4x2 + 9y2 = 36 and 4x2 + 9y2 = 72 have the same eccentricities.
89.Prove that the ellipses 16x2 + 9y2 = 144 and 17x2 + 10y2 = 170 have the same focuses.
90.Find intersection points of the parabola y2 = 12x and the ellipse
x2 + y2 = 1.
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91. For each of the following equations, show that the point M is on the ellipse and nd the equation of the tangent:
a) 3x2 + 8y2 = 25, M(1; 2), |
b) 5x2 + 2y2 = 98, M(4; 3); |
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c) x2 + 4y2 |
= 25, M(3; 2), |
d) 9y2 + x2 = 25, M(4; 1); |
e) 9x2 + y2 |
= 25, M( 1; 4), |
f) 6x2 + 11y2 = 98, M(3; 2). |
92. Find equations of the tangents to the ellipse 7x2 + 3y2 = 28 which have the slope 23, nd the intercepts and the tangency point of each tangent.
93. Find equations of the tangents through the point (8; 1) to the ellipse 2x2 + 5y2 = 70 and their tangency points.