Distance Formula :
222121dxxyy
Midpoint :
12mxxx2
12myyy2
General Equation of a Line:
AxByC0
2121yyriseyslopemtanrunxxx
AmB
Position of Line
Slope
Inclination
Inclined to the right
Positive
090
Inclined to the left
Negative
90180
Horizontal
Zero
0/180
vertical
Undefined/no slope
90
Four Standard Forms of a Line
1) Two-Point Form
211121yyyyxxxx
2) Point-Slope Form
11yymxx
3) Slope – Intercept Form
ymxb
4) Intercept Form
xy1ab
Distance from a Point to a Line
1122AxByCdAB
Angle between to Intersecting Lines
mm121mm12tan
Conic Sections
22AxBxyCyDxEyF0
Circle
2B4AC0,AC
Ellipse
2B4AC0,AC
Parabola
2B4AC0
Hyperbola
2B4AC0
when B=0
Circle
AC
Ellipse
AC
Parabola
A0
or
C0
Hyperbola A and C have opposite signs
Circle
Standard Form :
222xhykr
General Equation :
22AxCyDxEyF0
where
AC
Parabola
opening up or down
2AxDxEyF0
opening left or right
2CyDxEyF0
Position of Axis
Position of Parabola
Standard Form
Location of F and V
Vertical
Opens Upward
2xh4pyk
F is above V
Opens Downward
2xh4pyk
F is below the V
Horizontal
Opens to the right
2yk4pxh
F is at the right side of V
Opens to the left
2yk4pxh
F is at the left side of V
Ellipse
Position of Major Axis
Standard Form
General Equation
Horizontal
2222xhyk1ab
22AxCyDxEyF0
AC
Vertical
2222xhyk1ba
22AxCyDxEyF0
AC
Hyperbola
Position of Transverse Axis
Standard Form
General Equation
Horizontal
2222xhyk1ab
22AxCyDxEyF0
where A and C have opposite signs
Vertical
2222ykxh1ab
1. Which of the following is inclined to the right?
A.
3x50
C.
3x5y10
B.
3x5y10
D.
3y50
2. If a set of points have equal abscissas, then, they are lying on lines which are
A. inclined to the right C. inclined to the left
B. vertical D. horizontal
3. Which of the following has 0 inclination?
A. y = 6x + 0 C. y = 4
B. x = 4 D. y = 2x + 6
4. The inclination of the line 3x – 4y + 1 = 0
A. 36.87 C. 143.13
B. 53.13 D. 126.87
5. The line segment connecting
6,x
and
y,9
is bisected by the point
3,7
. Find the values of x and y.
A. 14,6 C. 33,12
B. 5,0 D. 6,14
6. Find the shortest distance from a point with coordinates
2,2
to the line
5x12y50
A. 3 C. 4
B. 5 D. 6
7. The inclination of L1 is arctan1/2. If L2 makes an angle of 45 with L1, find the slope of L2.
A. 3 C.
13
B.
13
D. – 3
8. Find the area of the triangle when the line
2x3y60
intersects with the coordinate axes.
A. 3 C. 4
B. 5 D. 2
9. The equation of the line that intercepts the x-axis at x = 4 and the y-axis at y = 6 is
A.
3x2y12
C.
2x3y12
B.
3x2y12
D.
2x3y12
10. Find the equation of the angle bisector of the acute angle formed between
3x2y50
and
4x6y10
.
A.
5x4y30
C.
2x2y90
B.
10x10y110
D.
10x8y110
11. Which of the following is a line at a distance 4 and parallel to
6x8y30
?
A.
6x8y430
C.
6x8y370
B.
6x8y430
D.
6x8y370
12. In a conic, the constant ratio between the set of points from its fixed point and fixed line is_
A. Intercept C. Inclination
B. Slope D. Eccentricity
13. What conic is formed when the cutting plane is parallel to an element?
A. Hyperbola C. Circle
B. Parabola D. Ellipse
14. The point on the parabola midway between its latus rectum and directrix
A. Vertex C. Center
B. Focus D. Latus rectum
15. If the point P is on the conic, F is the fixed point and D is the fixed line, and the distances between them are PF and PD; which statement is INCORRECT?
A. PF<PD but approaching one, the conic is an ellipse
B. PF=PD, the conic is a parabola
C. PF<PD but approaching zero, the conic approaches the form of a circle
D. PF>PD, no conic is formed
16. The parabola
2x8x8y0
is opening
A. To the Left C. To the Right
B. Upward D. Downward
17. The equation
22xy2y10
represents a
A. Circle C. Unit circle
B. Point D. Parabola
18. The standard form of the parabola facing downward with vertex at
2,4
and latus rectum is 5
A.
2y220x4
C.
2y25x4
B.
2x420y2
D.
2x45y2
19. How far from the y-axis is the center of the curve
222x2y10x6y550
?
A. 2.5 C. 3.0
B. 6 D. 1.5
20. Find the equation of the parabola passing through
2,2
, with vertex at
5,4
and directrix parallel to the y-axis.
A.
23y4x24y680
C.
23x30x4y910
B.
23x30x4y590
D.
23y4x24y280
21. The equation of a circle whose center is at
1,3
and passing through
2,1
.
A.
22xy6x2y70
C.
22xy6x2y70
B.
22xy6x2y70
D.
22xy6x2y70
22. The equation of the axis of the function
2y2x7x5
:
A. 7x + 4 = 0 C. 4x + 7 = 0
B. 4x – 7 = 0 D. x – 2 = 0
23. A chord passes through the focus of the parabola
2y4x
and the point
4,4
on the parabola. Determine the coordinates of the second point on the parabola where the chord passes through.
A.
11,4
C.
1,14
B.
11,4
D.
1,14
The general equation of an ellipse is
2225x16y50x64y3110
, find
25. its standard form
A.
22x1y211625
C.
22x1y211625
B.
22x1y211625
D.
22x1y211625
26. the coordinates of the center
A. (1,2) C. (1, 2)
B. (1, 2) D. (1,2)
27. the endpoints of the major axis of
A. (3,2) and (5,2) C. (1,3) and (1, 7)
B. (3,2) and (5,2) D. (1,7) and (1,3)
28. the coordinates of the foci
A. (2,2) and (4,2) C. (1,2) and (1,6)
B. (3,2) and (5,2) D. (1,1) and (1,5)
A hyperbola has its center at
1,4
, transverse axis parallel to the y – axis, distance between foci 10 and LR =
184
29. Find its vertices
A. (0,1) and (8,1) C. (1,0) and (1,8)
B. (3,4) and (6,4) D. (4,3) and (4,5)
30. Find the length of its conjugate axis
A. 3 C. 6
B. 9 D. 18
31. Find its standard form
A.
22y1x41916
C.
22x4y11916
B.
22y1x41169
D.
22x4y11169
32. Find its general equation
A.
2216x9y128x18y1030
B.
229x16y72x32y160
C.
2216x9y128x18y3910
D.
229x16y72x32y2720
33. Find the equation of its asymptotes
A.
4x3y190,4x3y130
B.
4x3y130,4x3y190
C.
3x4y160,3x4y80
D.
3x4y80,3x4y160
34. Find the equation of the circle that has its center at the focus of the parabola
2x2x8y230
and that is tangent to this parabola at its vertex.
A.
22xy2x2y20
B.
22xy2x2y40
C.
22xy2x2y0
D.
22xy2x2y20
35. Find the general equation of the ellipse which has the vertices of the hyperbola
225x4y20
as foci and the foci of the hyperbola as vertices.
A.
229x4y36
C.
225x4y20
B.
224x9y36
D.
224x5y20
