Analytic geometry. Textbook
.pdf4.2. Second order curves on the plane |
121 |
94. Find equations of those tangents to the ellipse 16x2 + 9y2 = 144 which have equal intercepts and nd their tangency points.
95. Show that the point M(2; 3) is on the ellipse x2 + y2 = 1 and nd
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the equation of the tangent to this ellipse at this point.
96. Find equations of the tangents to the following ellipses under the stated conditions and nd the points of tangency:
a)x2 + y2 = 1, the tangent passes through the point ( 6; 3);
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b)x2 + y2 = 1, the tangent is parallel to the line 2x y 17 = 0;
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97.2 |
Show that the line 4x 5y 40 = 0 is tangent to the ellipse |
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98. Find the equation of the tangent to the ellipse x2 + y2 = 1 for which
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the ratio of its distances from two focuses is 9.
99. The ellipse de ned by the standard equation passes through the point (3; 12=3) and is tangent to the straight line 4x + 5y 25 = 0. Find the equation of this ellipse and the tangency point.
100. Find the equation of the ellipse which is de ned by the standard equation and touches the lines x + y = 5 and x 4y = 10.
101. Find equations of common tangents for the following pairs of ellipses:
a) x2 |
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= 1 and x2 |
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= 1. |
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+ y2 = 1 and x2 |
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= 1. |
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102. Find common tangents of the ellipse x2 + y2 = 1 and the parabola
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y2 = 203 x.
103. Find focuses, eccentricity and directrices of each of the following hyperbolas:
a) 4x2 25y2 = 100; |
b) 9x2 4y2 |
= 36; |
c) 4y2 9x2 = 36; |
d) x2 y2 = 64; |
e) x2 y2 = 64; |
f) 3x2 8y2 = 48; |
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g) x2 4y2 = 4; |
h) 7x2 2y2 |
= 63. |
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Chapter 4. Problems |
104. Find standard equations of the hyperbolas satisfying the following conditions:
a)one vertex is (4; 0), and one focus is (5; 0);
b)one vertex is (0; 8), and the eccentricity is 2;
c)one asymptote is 2y = 3x, and one focus is (13; 0); p
d)the point (4; 3 ) is on the hyperbola, and (2; 0) is one vertex;
e)the points (4; 6) and (1; 1) are on the hyperbola;
f)one asymptote is 3x 4y = 0, and one vertex is (0; 10).
105.Find standard equations of hyperbolas with the following proper-
ties:
a)the distance between the vertices is 8, the distance between the focuses is 10;
b) the real semiaxis is 5, the vertices divide the distance between the
center and focuses in half;
c) the real semiaxis is 6, the hyperbola passes through the point (9; 4); |
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d) the hyperbola passes through the points ( 5; 2) and (2 5 ; |
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e) the points ( 10; 0) are the focuses, the hyperbola passes through the p
point (12; 3 5 );
f) the hyperbola has common focuses with the ellipse x2 + y2 = 1, and
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the eccentricity of the hyperbola e = 1:25;
g) the hyperbola passes through the focuses of the ellipse x2 + y2 = 1
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and has the focuses at the vertices of this ellipse.
106.Find the semiaxes of the hyperbolas with the following properties:
a)the distance between the focuses is 8, the distance between the direc-
trices is 6;
p
b)the directrices are de ned by the equations x = 3 2 , the asymptotes form a right angle;
c)the asymptotes are de ned by the equations y = 2x and the focuses are at a distance of 5 units from the center;
d)the asymptotes are de ned by the equations y = 53x, the hyperbola passes through the point (6; 9).
4.2. Second order curves on the plane |
123 |
107. For the hyperbola x92 16y2 = 1:
a) nd coordinates of the focuses; b) nd the eccentricity;
c) write equations of the asymptotes and directrices.
108. Find the standard equation of the hyperbola which has the asymp- |
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totes y = |
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2x and passes through the point (12; 3 |
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109. Find the angle between the asymptotes of the hyperbolas with the following properties:
a)the eccentricity e = 2;
b)the distance between the focuses is twice the distance between the directrices.
110.On the hyperbola x252 24y2 = 1, a point is taken, whose abscissa is 10, and the ordinate is positive. Find the focal distances of this point.
111. Find equations of the straight lines which are parallel to the asymptotes of the hyperbola x2 4y2 = 4 and pass through the point (2; 5).
112. In each of the following cases show that the point M is on the
given hyperbola, and nd the equation of the tangent at this point:
a) 4x2 y2 = 64, M(5; 6); |
b) x2 9y2 = 25, M( 13; 4); |
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113. Find the equations of the straight lines which are tangent to the hyperbola 16x2 25y2 = 400 and parallel to the straight line 2x 2y = 5,nd the corresponding tangency points.
114. Find the equations of tangents to the hyperbola 4x2 3y2 = 96 which are parallel to the straight line y = 2x, and nd the corresponding tangency points.
124 Chapter 4. Problems
115. Find equations of tangents to the hyperbola which pass through
the point M, and nd the corresponding tangency points: a) 5x2 2y2 = 18, M(1; 4); b) x2 y2 = 9, M(3; 9).
116.Find equations of tangents to the hyperbola x2 4y2 = 16 which have equal intercepts on the axes, and nd the corresponding tangency points.
117.Find equations of the tangents to the hyperbola x82 y92 = 1 and passing through each of the following points a) (2; 0); b) ( 4; 3); c) (5; 1).
118.Find the tangents to the hyperbola x82 y92 = 1 which are:
a)parallel to the line x + y 7 = 0.
b)perpendicular to the line x 2y = 0.
119.Find the standard equation of the hyperbola with the tangent x y 2 = 0 at the tangency point (4; 2).
4.3Analytic geometry in the space
120.Find the lengths of the segments with the ends at the following pairs of points:
a) (4; 3; 5), (1; 3; 1); |
b) ( 5; 4; 1), (1; 2; 4); |
c) (2; 4; 1), ( 2; 0; 3); d) (3; 5; 2), (1; 1; 5);
e)(4; 5; 2), ( 3; 1; 2).
121.Find the equation of the plane passing through the point
M0(2; 3; 4) and parallel to the plane x 3y + 4z + 14 = 0.
122. Find the common point of the planes x+y+z 4 = 0, x y z = 0, and 3x y + z 6 = 0.
123. Find the equation of the plane which is parallel to the plane 3x 6y 2z + 14 = 0 and is 3 units from this plane.
4.3. Analytic geometry in the space |
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124. Find the intercepts of the following planes: |
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a) 2x 3y z + 12 = 0; |
b) 5x + y 3z 15 = 0; |
c) x y + z 1 = 0; |
d) x 4y + 6 = 0; |
e) 5x 2y + z = 0; |
f) x 7 = 0. |
125. Change the following equations of planes to the normal form: a) 2x 9y + 6z 22 = 0; b) 10x + 2y 11z + 60 = 0;
c) 6x 6y 7z + 33 = 0.
126.Find the distance of the plane 15x 10y + 6z 190 = 0 from the
origin.
127.Find equations of the planes under the following conditions:
a)passing through the point ( 2; 7; 3) and parallel to the plane x 4y + 5z 1 = 0;
b) passing through the origin and orthogonal to the planes 2x y + 5z +
3 = 0 and x + 3y z 7 = 0.
128.Find the distance of the plane 15x 10y + 6z 190 = 0 from the
origin.
129.On the z-axis nd a point equidistant from the planes x + 4y 3z 2 = 0 and 5x + z + 8 = 0:
130.Find the distance between the planes
11x 2y 10z + 15 = 0 and 11x 2y 10z 45 = 0:
131.Find the equation of the plane which is parallel to the plane 3x 6y 2z + 14 = 0 and is 3 units from this plane.
132.Find the equation of the plane passing through the origin and the points (3; 2; 1) and (1; 4; 0).
133.Find the equation of the plane that is 3 units from the plane 3x + 6y + 2z + 4 = 0.
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Chapter 4. Problems |
134. Find the equation of the plane passing through the intersection line of the planes 4x y + 3z 1 = 0 and x + 5y z + 2 = 0 under the stated conditions:
a)passing through the origin;
b)passing through the point (1; 1; 1);
c)parallel to the y-axis;
d)perpendicular to the plane 2x y + 5z 3 = 0.
135.Find the equation of the plane passing through the point (3; 2; 4) and having equal intercepts.
136.Find the equation of the plane:
a)parallel to the plane yOz and passing through the point (1; 3; 2);
b)passing through the x-axis and the point ( 3; 1; 4);
c)parallel to the y-axis and passing through the points (6; 2; 0) and (7; 3; 9).
137.Find the intercepts of the plane 5x + y 3z 15 = 0.
138.Find the common point of the planes x+y+z 4 = 0, x y z = 0, and 3x y + z 6 = 0.
139.Find the equation of the plane passing through the origin and the points (3; 2; 1) and (1; 4; 0).
140.Find equations of the straight line passing through the points (2; 3; 1) and (1; 0; 5).
141.Find equations of the straight line passing through the point (2; 1; 1) and parallel to the straight line
( 2x |
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142. Find equations of the straight line passing through the point |
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(2; 4; 3) and parallel to the straight line |
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4.3. Analytic geometry in the space |
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143. Find canonical equations of the straight line
(
2x 3y 3z 9 = 0; x 2y + z 3 = 0:
144. Find coordinates of each of the points in which the straight line
(
3x + y 3z = 10;
2x + 2y z = 4
cuts the three coordinate planes.
145. Find the distance of the point (7; 9; 7) from the straight line
x 2 = y 1 = z :
4 3 2
146. Find the point on the straight line
x = y + 7 = z 3 1 2 1
that is nearest to the point (3; 2; 6).
147. Find the point on the straight line
(
x + 2y + z 1 = 0;
3x y + 4z 29 = 0
which is equidistant from the points (3; 11; 4) and ( 5; 13; 2).
148. Find the distance between two parallel straight lines
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149. Find the distance between the following pairs of skew straight lines:
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Chapter 4. Problems |
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150. Check whether the following lines intersect, and if they intersect, |
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nd the intersection point: |
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151. Find equations of the perpendicular dropped from the origin to |
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the line |
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152. Find equations of the line passing through the point (4; 0; 1) and |
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intersecting the straight lines |
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153. Find equations of the straight line intersecting the lines |
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154. Find equations of the common perpendicular to the lines |
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155. Find the equation of the straight line passing through the point |
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(2; 5; 3) under the following conditions: |
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5x + 4y |
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4.3. Analytic geometry in the space |
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129 |
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156. Find the intersection points of the straight line |
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with the sphere x2 + y2
157.Find equations of the planes under the following conditions:
a)orthogonal to the straight line
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and 5 units from the origin. |
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b) passing through the point (4; 3; 6) and perpendicular to the line
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c) passing through the point (1; 2; 0) and orthogonal to the straight line which passes through the points (2; 4; 2) and ( 1; 3; 7).
d) passing through the point (2; 3; 7) and having equal intercepts on the axes.
e) parallel to the plane 2x 6y + 3z 5 = 0 and 6 units from the origin.
158. Find equations of the straight line passing through the point (3; 2; 1) and orthogonal to the plane 4x + y 6z + 11 = 0.
159. Find equations of the straight line passing through the origin and orthogonal to the plane 3x 2y + 5z 12 = 0.
160. Find coordinates of the intersection point of the plane 6x 8y + 2z 4 = 0 and the line passing through the points (2; 2; 4) and (6; 0; 2).
161. Find the equation of the plane that contains the straight line p
x = y = z 6 and is 6 units from the origin.
162. Find the equation of the plane passing through the intersection line of the planes
x + 28y 2z + 17 = 0 and 5x + 8y z + 1 = 0
and tangent to the sphere x2 + y2 + z2 = 1.
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Chapter 4. Problems |
163. Find the intersection point of the straight line and the plane |
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b) |
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c) |
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164. Find canonical equations of the straight line passing through the
intersection points of the plane 2x + y 3z + 1 = 0 with the straight lines
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165. Find the value of the coe cient A so that the plane Ax + 3y |
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5z + 1 = 0 is parallel to the straight line |
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166. Find the values of the coe cients A and B so that the plane
Ax + By + 6z 7 = 0 is orthogonal to the straight line
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167. Find equation of the plane passing through the origin and orthog- |
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onal to the straight line |
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168. Find equations of the straight line passing through the point (1; 0; 7), parallel to the plane 3x y + 2z 15 = 0 and intersecting the
straight line
x 1 = y 3 = z :
4 2 1
169. For each pair consisting of a line and a plane, determine whether:
• the line lies on the plane;