ANALYTIC GEOMETRY 2 LECTURE

10. Which of the following is the equation of a circle

1. What type of conic is represented by x^2-
with origin as center and passing through the vertices
4xy+3y+5=0?
of an equilateral triangle whose median is of length
A. Parabola
B. Hyperbola 3a?
C. Ellipse A. 𝑥 2 + 𝑦 2 = 9𝑎2
D. Circle B. 𝑥 2 + 𝑦 2 = 4𝑎2
C. 𝑥 2 + 𝑦 2 = 16𝑎2
2. What conic section is described by the given D. 𝑥 2 + 𝑦 2 = 3𝑎2
equation? 11. Which of the following statement is FALSE for all
4𝑥 2 − 𝑦 2 + 8𝑥 + 4𝑦 = 15 noncircular ellipses?
A. Parabola Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
B. Hyperbola A. The eccentricity e is less than one.
C. Ellipse B. The ellipse has two foci.
D. Circle
C. The sum of the two distances from the two foci to
3. The equation describes: x^2+y^2-4x+2y-20=0
any point on the ellipse is 2a (i.e. twice the semi
A. A circle of radius 5 centered at the origin
B. An ellipse centered at (2,-1) major distance).
C. A sphere centered at the origin D. The coefficients A and C preceding the 𝑥 2 and 𝑦 2
D. A circle radius 5 centered at (2,-1) terms in the general form of the equation are equal.
4. Find the farthest distance from the point (12, 2) to the 12. Find the center and major axis of the ellipse:
circle x^2+y^2+6x-16y+24=0 : 𝑥 2 + 4𝑦 2 − 2𝑥 − 8𝑦 + 1 = 0
A. 7 A. (1, 1), 4
B. 6.156 B. (2, -4), 1
C. 23.156 C. (1, -1), 2
D. 21.156 D. (-1, 1), ½
5. The length of the tangent from (4,8) to the circle 13.
𝑥 2 + (𝑦 − 1)2 = 32 is:
A. 3.81
B. 4.14
C. 5.66
D. 7.48 14. Determine the coordinates of the foci of the
6. What is the equation of the circle passing through the equation:
(x,y) points (0,0), (0,4), and (-4,0)?
A. (𝑥 − 2)2 + (𝑦 − 2)2 = √8 25𝑥 2 + 16𝑦 2 − 150𝑥 + 128𝑦 + 81 = 0
B. (𝑥 − 2)2 + (𝑦 − 2)2 = 8 A. (3, -1) and (3, -7)
C. (𝑥 + 2)2 + (𝑦 − 2)2 = 8 B. (4, -2) and (5, -3)
D. (𝑥 + 2)2 + (𝑦 + 2)2 = √8 C. (3, -4) and (4, -3)
7. Which of the following points (1, 0), (-1, 0), (4, 4) D. (2, -4) and (3, -2)
and (9, 7) belong to the equation y = 𝑥 2 − 𝑥? 15. A satellite orbits around the earth in an elliptical path
A. (-1, 0) of eccentricity 0.6 and semi-minor axis of length
B. (9, 7) 12,000 miles. If the center of the earth is at one of the
C. (4, 4) and (1, 0) foci, find the maximum altitude of the satellite.
D. (1, 0) A. 24,000 miles
8. B. 31,000 miles
C. 36,000 miles
D. 27,000 miles
9. The x- and y-coordinates of a particle moving in the 16. Determine the length of the latus rectum of the
x-y plane are x = 8sint and y = 6cost. Which of the hyperbola whose equation is: 9𝑥 2 − 4𝑦 2 + 90𝑥 +
following equation describes the path of the particle? 189 = 0
A. 36𝑥 2 + 64𝑦 2 = 2304 A. 9
B. 4
B. 6𝑥 2 + 8𝑦 2 = 10
C. 3
C. 64𝑥 2 + 36𝑦 2 = 2304
D. 9/4
D. 64𝑥 2 − 36𝑦 2 = 2304 17. Find the location of the vertex of the hyperbola:
16𝑥 2 − 9𝑦 2 + 32𝑥 + 36𝑦 − 36 = 0
A. (-2,2), (0,2)
ANALYTIC GEOMETRY 2 LECTURE
B. (1,2), (2,4) 26. The cables of a suspension bridge are in the shape of
C. (2,0), (2,-6) a parabola. The towers supporting the cable are 600 ft
D. (2,2), (2,-4) apart and 80 ft high. If the cables touch the road
18. Find the equation of the asymptote of the hyperbola surface midway between towers, what is the height of
𝑥2 𝑦2 the cable from the road at a point 150 feet from the
− =1
9 4 center of the bridge?
A. 2x-3y=0 A. 10 ft
B. 3x-2y=0 B. 20 ft
C. 2x-y=0 C. 30 ft
D. 2x+y=0 D. 40 ft
19. Given the equation of a parabola 27. When the load is uniformly distributed horizontally,
𝑦 2 + 3𝑥 − 4𝑦 + 7 = 0 the cable of a suspension bridge hangs in a parabolic
Determine the following: arc. If the bridge is 300 ft long, the towers 60 ft high
i.) Length of the latus rectum and the cable is 20 ft above the roadbed at the center,
ii.) Vertex find the vertical distance from the roadbed 50 ft from
iii.) Focus the center.
20. Given the equation of a parabola A. 10.44 ft
𝑥 2 − 4𝑥 − 8𝑦 − 6 = 0 B. 24.44 ft
Determine the following: C. 30.44 ft
i.) Length of the latus rectum D. 40.44 ft
ii.) Vertex 28. The polar coordinates (r,θ) of a point are (4, 120°).
iii.) Focus What are the rectangular (x, y) coordinates?
21. Determine the equation of a parabola with vertex (-2, A. (3.46, 2)
3) and focus at (-4, 3) B. (2, 3.46)
A. 𝑦 2 + 8𝑥 − 6𝑦 + 25 = 0 C. (-2, 3.46)
B. 𝑥 2 + 9𝑥 − 8𝑦 − 20 = 0 D. (-2, -3.46)
C. 𝑦 2 − 5𝑥 − 7𝑦 + 40 = 0 29. What is the equivalent Cartesian equation of the
𝜋
D. 𝑥 2 + 9𝑥 − 9𝑦 + 24 = 0 given 𝜃 = ?
3
22. The location of the focus of the parabola: 4𝑦 2 + A. 𝑦 = √3𝑥
8𝑥 − 12𝑦 − 7 = 0 𝑖𝑠 𝑎𝑡::
B. 𝑥 = √3𝑦
A. (0, 3/2)
B. (2, -3/2) C. 3𝑦 = √3𝑥
C. (3/2, 3/2) D. 5𝑦 = √3𝑥
D. (-2, -3/2) 30. What is the polar equation of a circle of radius 3 units
23. Determine the equation of a parabola with focus at and center at (3, 0).
(4, 2) and directrix of y = -4. A. r = 6cos𝜃
A. 𝑥 2 − 8𝑥 − 12𝑦 + 4 = 0 B. 𝑟 2 = 4𝑠𝑖𝑛𝜃
B. 𝑥 2 − 10𝑥 − 27𝑦 − 2 = 0 C. r = 4cos𝜃
C. 𝑦 2 − 2𝑥 − 7𝑦 − 8 = 0 D. r = 10sin𝜃
D. 𝑦 2 − 6𝑥 − 2𝑦 − 24 = 0 31.
24. What is the harmonic mean of the segments of a focal
chord of the parabola y(squared) = 16ax?
A. 2a
B. 4a
C. 8a 32.
D. ½ a
25. Engr. Jomar Galindo with his wife Dahyun is driving
a truck hauling a cylindrical tank has to pass under
overhead parabolic arch bridge which has 24 m wide
base and is 20 m high. If the tank is 16 m in diameter
and is placed in the truck with sides vertical and its
top 10m above the ground. Find the smallest
clearance from the top of the tank.
A. 8.49
B. 1.11
C. 11.1
D. 0.49