Analytic Geometry and Mark Anthony T.

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1906 - 2006
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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