RASCO CIVIL ENGINEERING REVIEW

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ANALYTIC GEOMETRY

DISTANCE FORMULA
21. Find the distance of the two parallel lines x + 4y = -6 and 2x + 8y
1. Find the distance between P1 (9, 2) and P2 (5,5). = 22.

2. Find the midpoint between P1 (9, 6) and P2 (3,-2). 22. Find the smallest angle between the lines 2x + y – 8 = 0 and x +
3y + 4 = 0.
3. The distance between the points (1, 3) and (9, k) is 8. Find the
value of k. 23. Find the equation of the bisector of the obtuse angle between
the lines 3x + 4y + 3 = 0 and the line 4x -3y + 12=0.
DIVISION OF LINE SEGMENT
CIRCLE
4. Determine the point of division of the line segment from A (5, 6)
to B (-3, -2) that divides this line segment, starting from A, into two 24. Compute the area of the circle having an equation of x^2 + y^2 -
parts in the ratio 1 : 4. 10x + 4y – 196 = 0.

5. The segment from (-2, 6) to (4, -4) is extended 3 times its own 25. What is the equation of the circle whose center is at (5, 5) with a
length. Find the terminal point. radius of 5?

AREA OF POLYGON 26. Find the equation of the radical axis of two circles having an
equation of x^2 + y^2 + 4x + 6y -3 = 0 and x^2 + y^2 + 12x + 14y + 60
6. Find the area of the polygon bounded by (-3, 5), (1, -2), (-1, -2) and =0
(6, 4).
27. Compute the length of the chord of a circle having an equation
7. The area of the triangle whose vertices is at (5, 1), (-3, 1), and (1, of x^2 + y^2 -16x = 0 if the distance from the chord to the center of
y) is 24. Find the value of y. the circle is 4 units.

CENTROID OF TRIANGLE 28. Find the equation of the circle that passes through the points (1,
-2), (5, 4), and (10, 5)
8. Find the coordinates of the centroid of the triangle ABC with
coordinates of A(0, -3), B(3, 0), and C(0, -6). PARABOLA

LOCUS OF POINTS Given the parabola x^2 – 6x – 12y – 51 =0

Find the equation of the locus of the following conditions: 29. determine the focus

9. Equidistant from (3, -2), (4, 3) 30. determine the ends of latus rectum

10. Its distance from (4, 0) is twice its distance from the line x = 1. 31. determine the equation of the directrix

11. Its distance from (6, 0) is always thrice its distance from (1, 0) 32. Find the equation of a parabola with axis parallel to x and
passing through (-10, 11), (-4, 7), (-2, 3).
12. Equidistant from y = 5 and point (0, -5)
33. Find the equation of the directrix of the parabola having its
13. the sum of its distance from (3, 0) and (-3, 0) is 10 vertex at (6, 0) and passing through (2, 1) whose axis is parallel to
the y-axis.
14. the difference of its distance from (0, 4) and (0, -4) is 6.
34. Find the distance from the vertex of a parabola, x^2 = 4(y-2) to
15. The two vertices of a triangle are (2, 4) and (-2, 3) and the area is the line 3x + 4y + 2=0.
2 sq units. Compute the locus of the 3rd vertex.
35. An arch 18 m high has the form of a parabola with vertical axis.
LINES The length of a horizontal beam placed across the arch 8 m from the
top is 64 m. Find the width of the arch at the bottom.
16. Determine the equation of the line whose slope is 3/4 and
passes through (2, 3) 36. If an automobile head light reflector is cut by a plane through its
axis, the section is a parabola having the light center as a focus. If
17. Determine the equation of line that passes through (-4, 3) and (2, the light is 18 mm from the vertex and the diameter of the light is
5) 250 mm, find the depth of the head light.

Find the equation of the line that passes through (4, 2) 37. A fixed circle in the 1st quadrant has its center at (6, 8) and a
radius of 4. Find the locus of a moving circle if it is always tangent to
18. parallel to line 3x – 4y = 7 the x-axis and the fixed circle.

19. perpendicular to a line 2x + 6y = 9 ELLIPSE

20. Find the distance from 9x = 8y – 12 to point (6, -1) Given the ellipse 4x^2 + 9y^2 – 24x - 72y + 144 = 0, determine
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 RASCO CIVIL ENGINEERING REVIEW
Aloja Coffee Shop, Libas, Banga, Aklan
CP: 09164618841 / Landline: (036) 267 – 7343

ANALYTIC GEOMETRY

58. Find the equation of the hyperbola whose center is (2, 3), whose
38. center distance of latus rectum is 4.5, distance between foci is 10, with
eccentricity of 1.25, and transverse axis parallel to x.
39. vertices
TRANSLATION OF AXES
40. equation of the directrix
59. Find the new coordinate of the point P(4, -2) if the origin is
41. Latus rectum moved to (-2, 3) by a translation.

42. eccentricity TRANSFORMATION

43. Consider the center of the ellipse is the origin, find the equation 60. Transform the equation 2xy + 9 =0 to remove the xy term.
of ellipse whose eccentricity is 2/3, distance between foci is 2 and
parallel to x. 61. Find the acute angle of rotation such that the transformation
equation of 2x^2 + sqrt3 xy + y^2 = 8 will have no xy term.
44. Consider the center of the ellipse is at (-1, 3), find the equation
of ellipse whose eccentricity is 1/2, distance between directrices is POLAR COORDINATES
24 and parallel to y-axis.
62. Determine the polar form of the rectangular point P (8, 14).
45. An ellipse is a locus of a point so that the sum of its distances
from two fixed points (0, 3) and (0, -3) is always equal to 8. Compute 63. Determine the rectangular form of the polar point P (4, 2pi/3).
the distance from the center to the directrix, its eccentricity, and
length of latus rectum. 64. The vertices of a triangle have polar coordinates of (0, 0 deg), (6,
30 deg), and (9, 70 deg). Find the area and perimeter of the triangle.
46. An ellipse has a general equation of Ax^2 + By^2 + F =0 passing
thru (8, 0) and (0, 6). Find the value of A, B, and its area. 65. Identify the polar curves described by

47. The earth’s orbit is an ellipse with the sun as one of the foci. If a. r = 4cos 3
the farthest distance of the sun from the earth is 105.5 million km, b. r = 6sin 
and the nearest distance of the sun from the earth is 78.25 million
km, find the eccentricity of the ellipse. 66. Determine the length of the latus rectum of the curve r = 4 / (1 –
sin theta).
48. An ellipse has a length of semi-major axis of 500 cm and a semi-
minor axis of 300 cm. Compute its second eccentricity. 67. Find the total length of the curve r = 2 (sin theta + cos theta)

49. The perimeter of an ellipse is 28.448 cm. The major axis is 10 cm 68. Write the equivalent polar equation of the line 3x + 4y = 20.
long lies on the x-axis with its center at the origin. Determine its
equation. COORDINATES IN SPACE

HYPERBOLA 69. Find the polar coordinates of a point P having rectangular

coordinates of (3, 4, 5)?
Given the hyperbola 9y^2 - 4x^2 – 18y + 24x – 63 = 0.
70. What is the equivalent spherical coordinates of a point P having
50. determine the ends of transverse axis rectangular coordinates of (3, 4, 5)?

51. determine the ends of conjugate axis 71. What is the equivalent cylindrical coordinates of a point P having
rectangular coordinates of (3, 4, 5)?
52. determine the equation of asymptotes

A hyperbola has an equation of 16x^2 – 9y^2 -128x -90y -113 = 0.

Compute PROBLEM SETS

53. Eccentricity 1. It is a pattern of point that describes its relation to distance of a

point (x, y) from one fixed entity to that of another entity.
54. Distance of directrix from the center of curve a. function c. scheme
b. locus d. diagram
55. Length of latus rectum
2. Which of the following statement is always true?
56. A point moves so that the difference between its distance from a. the slope of the line is tangent of the angle of inclination of the
(0, 5) and (0, -5) is always 8. Find the equation of the locus. line from the positive x-axis.
b. if two slope of lines have the same absolute value but opposite of
57. What is the equation of the asymptote of the hyperbola x^2/9 – sign, the lines are always perpendicular
y^2/4 = 1. c. if one line is multiple of another line, the two lines are parallel
d. none of the above.

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 RASCO CIVIL ENGINEERING REVIEW
Aloja Coffee Shop, Libas, Banga, Aklan
CP: 09164618841 / Landline: (036) 267 – 7343

ANALYTIC GEOMETRY

3. Plane figures that can be obtained by the intersection of a double a. 32 c. 16

cone with a plane passing through the apex. These includes a point, b. 20 d. 10
a line, and intersecting lines
a. conic sections 16. length of minor axis
b. generate conic sections a. 32 c. 16
c. degenerate conic sections b. 20 d. 10
d. great conic sections
17. length of its latus rectum
4. Find the distance between the given lines 4x–3y = 18 and 8x – 6y = a. 12.5 c. 128
-30. b. 51.2 d. 16
a. 6.6 c. 0.6
b. -6.6 d. -0.6 18. how far apart are the directrices
a. 41 d. 32
5. Determine the equation of the lines through (-5, -3) and passing at b. 20.5 c. 16
distance 2sqrt5 from (5, 7)?
a. 6x – 5y = -15 c. 3x-4y +15=0 19. eccentricity of the ellipse
b. 2x – y + 7 = 0 d. 2y-4x =-7 a. 0.78 c. 0.8
b. 1.25 d. 1.28
6. CD is the diameter of a circle whose center is P. If the coordinates
of C is (2, 6), of D is (6, 20), find the coordinate of P. 20. second eccentricity of the ellipse
a. (4, -4) c. (4, 13) a. 0.78 c. 0.8
b. (4, 4) d. (0, 0) b. 1.25 d. 1.28

7. The equation of the ellipse with vertices at (-3, -2) and (1, -2) and 21. flatness of the ellipse
which passes through (-2, -1) is a. 0.375 c. 0.22
a. x^2 + 3y^2 + 2x + 12y + 9 =0 b. 0.6 d. 0.48
b. 3x^2 + y^2 + 4x + 15y + 9 =0
c. x^2 + y^2 - 14x + 12y - 12 =0 22. equation of the ellipse if center is at the origin and major axis
d. 3x^2 + 4y^2 - 12x + 3y - 9 =0 is parallel to x-axis
a. 25x^2 + 64y^2 = 6400 c. 100x^2 + 125y^2 = 12600
8. The vertices of a triangle are (4, 2), (4, 10), and (8, 2). Its area is b. 64x^2 + 25y^2 = 6400 d. 36x^2 + 25y^2 = 100
a. 12 c. 7
b. 10 d. 16 23. Determine equation of a straight line perpendicular to 2x – y – 3
= 0 and passes through (6, -3)
9. Find the equation of a straight line through the point (5, -4) and a. x + 2y = 0 c. x – 2y = 12
parallel to the line y = 3x -2 . b. y = -6 + 0.5x d. 2x – y - 15 = 0
a. 3x-y-19=0 c. 4x-3y=-4
b. 6x-5y=50 d. 2x-8y=5 From the polar curve r = 4/(4 + 2cos theta)

Given the equation x^2 +y^2 - 6x-8y-11=0 24. Determine the conic section being described
a. circle c. parabola
10. Determine the conic described. b. ellipse d. hyperbola
a. circle c. hyperbola
b. ellipse d. parabola 25. Determine its area
a. 4.84 c. 8.56
11. identity the center b. 3.77 d. 6pi
a. (-3, -4) c. (-3, 4)
b. (3, 4) d. (3, -4)

12. Solve the radius

a. 6 c. -6
b. 9 d. 11

13. determine the perimeter

a. 12pi c. -6pi
b. 36pi d. 18 pi

14. Determine the area

a. 12pi c. -6pi
b. 36pi d. 18 pi

The area of an ellipse is 502.655 sq units and its perimeter is 83.828

units. Determine

15. length of major axis

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