# Analytic Geometry and Mark Anthony T.

Republic of the Philippines

CAVITE STATE UNIVERSITY

(CvSU)

DON SEVERINO DE LAS ALAS CAMPUS

Indang, Cavite

(046) 415-0013 / (046) 415-0012

E-mail: cvsu@asia.com

Problem Set

(Families of Curves)

Submitted by:

BSCOE 3-1

Submitted to:

Engr. Jaykie Homer P. Hernandez

Garces, Johhn Rommel T.

6. Straight lines at fixed distance p from the origin

Ax+By+C=0

C=-Ax-By

A+By'=0

A=-By' where x,y=(0,0)

P=Ax+By+CA2+B2+C2

P=CA2+B2+C2

PA2+B2+C2=C

PA2+B2+C2=-Ax-By

P-By'2+B22=--By'x-By

PB2y'2+1=Bxy'-y

PB1+y'2=B(xy'-y)

P1+y'2=xy'-y2

P21+y'2=xy'-y2

xy'-y2=P21+y'2

Diaz, Mark Kenneth

7. Circles with center at the origin.

x-h2+y-k2=r2 ;Cb,0

x2+y2=r2

2x+2yy'2=0

x+yy'=0

x+ydydx=0

xdx+ydy=0

Creus, Ares Michael

8. Circles with center on the x-axis.

(x-h)2+(y-k)2=r2 ;C(h,0)

(x-h)2+y2=r2

2x-h1+(2yy')2=0

x-h+2yy'=0

1-0+y'y'+yy''=0

1+(y')2+yy''=0

Pernito, Edelyn

9. Circles with fixed radius r and tangent to the x-axis. x±h2+y±k2=r2 ;r=k

x±h2+y±k2=r2 equation 1

2x±h+2y'y±k2=0

x±h+y'y±k=0

x±h=-y'y±k equation 2

-y'y±r2+y±r2=r2

y'2y+r2+y2±2ry+r2=r2

y'2y+r2+y²±2ry=0

Flores, Bernadette L.

10. Circles tangent to the x-axis.

x-h2+y-r2=r2

x-h=r2-y-r2

1=-2y-ry'2r2-y-r2

r2-y-r2=--2y-ry'

x-h=-y'y-r

-y'y-r2+y-r2=r2

-y'2y-r2+y-r2=r2

y-r2y'2+1=r2

y-ry'2+1=r

yy'2+1-ry'2+1=r

yy'2+1=r+ry'2+1

yy'2+1=r1+y'2+1

yy'2+11+y'2+1=r

1+y'2+1y'y'2+1+yy'y''y'2+1-yy'2+1-yy'y''y'2+11+y'2+12=01+y'2+12 y'y'2+1=y'y'2+1+yy'y''y'2+1-yy'y''+yy'y''=0

y'y'2+1=y'y'2+1+yy'y''y'2+1=0y'2+1

y'y'2+1-y'3y'2+1-y'y'2+1+yy'y''y'=0

y'2+1+yy''=y'2y'2+1+y'2+1

y'2+1+yy''=y'2y'2+1+y'2+12

y'2+1+yy''2=y'2+12y'2+1

y'2+1+yy''2=y'2+13

Diaz, Mark Kenneth

11. Circles with the center on the line y=-x, and passing through the origin. r=h-x12+h-y12

r=h2+h2

r=2h2 ;r2=2h2

x-h2+y+h2=2h2 ;±h=∓k

x2-2xh+h2+y2+2yh+h2=2h2

2h=x2+y2y-x

y-x2x+2yy'-y'-1x2+y2y-x2=0y-x2

2xy+2y2y'-2x2-2xyy'-x2y'-y2y'+x2+y2=0

2xy+y2y'-x2-2xyy'-x2y'+y2=0

-x2+2xy+y2-x2y'-2xyy'+y2y'=0

--x2-2xy-y2-x2+2xy-y2y'=0-1

x2-2xy-y2+x2+2xy-y2dydx=0

x2-2xy-y2dx+x2+2xy-y2dy=0

Regaya, Jayson

12. Circles of radius unity. Use the fact that the radius of curvature is 1. x-h2+y-k2=0 equation 1

2x-h+2y-ky'2=0

x-h+y-ky'=0 equation 2

1+y'y'+y-ky''=0

1+y'2+y''y-k=0 equation 3

from equation 1

(x-h)2=1-(y-k)2

x-h=1-(y-k)2

from equation 2

1-(y-k)2+y-ky'=0

1-(y-k)2=-y-ky'

1-(y-k)2=(y-k)2(y')2(y-k)2

1(y-k)2=1+(y')2

(y-k)2=11+(y')2

y-k=1+(y')21+(y')2

from equation 3

1+y'2+y''1+(y')21+(y')2=01+y'2

1+y'22+y''1+(y')2=0

1+y'22=-y''1+(y')22

1+y'24=y''21+y'2

1+y'241+y'2=(y'')2

1+y'22=y''2

.

Diaz, Mary Nielby

13. All circles. Use the curvature.

x-h2+y-k2=r2

2x-h+2y-ky'2=0

x-h+y-ky'=0

1+y'2+y-ky''=0

1+y'2y''=-y-ky''y''

1+y'2y''=k-y

1+y'2y''+y=k

y''2y'y''-1+(y')2y'''+y'(y'')2=0

2y'y''3-y'''-y'3y'''+y'y''2=0

3y'(y'')2=y'''1+y'2

Pernito, Edelyn

14. Parabolas with vertex on the x-axis, with axis parallel to the y-axis, and with distance from focus to vertex fixed as a. x-h2=4ay-k;Ch,0

(x-h)2=4ay

2x-h=4ay'2

x-h=2ay'

4ay2=2ay'2

4ay=4ay2y'24a

y=a(y')2

Cabaluna, Mark Anthony T.

15. Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from focus to vertex fixed as a. (y-k)2=4ax-h;C(0,k)

(y-k)2=4ax

2y-ky'=4a2

y-ky'=2a

y'4ax2=2a2

4ax(y')2=4a4a

x(y')2=a

Senorio, Janine Joy

16. Parabolas with axis parallel to the y-axis and with distance from vertex to focus fixed as a. x-h2=4ay-k

2x-h2=4ay'2

x-h=2ay'

1=2ay''

Creus, Ares Michael

17. Parabolas with axis parallel to the x-axis and with distance from vertex to focus fixed as a. y-k2=4ax-h

2y-ky'=4a2

y-ky'=2a

y-k=2ay'

y'=y'0-2ay''y'2y'2

y'3=-2ay''

y'3+2ay''=0

Regaya, Jayson

18. Work Exercise 17, using differentiation with respect to y. y-k2=4ax-h

2y-k=4ax'2

y-k+2ax'

1=2ax''

2ax''=1

2ad2xdy2=1

.

Tapia, Cris John F.

19. Use the fact that...

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