0 Analytics Geometry Formula and Question and Answer Practice

 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