Math 165 M2
Your Name Below: (MUST BE WRITTEN CLEARLY)
October 2, 2008
HW 4
(You must show your work fully completely to gain the 4 extra credits.)
Due on Oct 2nd Thursday, No late HW will be accepted!
1. (Extra points: 3)
Normal line is the line perpendicular to the tangent line. Find the normal line to the curve
8(x2+y2)2= 100(x2−y2).
Solution: For given point x0, the corresponding y-coordinate is by solving
x4
0+ 2x2
0y2+y4=25
2(x2
0−y2)
so that
y2=1
2−(2x2
0+25
2) + 10rx2
0+25
16.
Note, from the above solution, one has x∈h−r50
4,r50
4i. So,
y0=±s−(x2
0+25
4)+5rx2
0+25
16.
Now, it is sufficiently to compute the slope by differentiating both sides of the equation is
m=−1
f0(x0)=25y+ 4(x2+y2)y
4(x2+y2)x−25x
so that the equation of normal line is
y=m(x−x0) + y0
that
y=25y0+ 4(x2
0+y2
0)y0
4(x2
0+y2
0)x0−25x0x−x0+y0
where x0is given and y0is given by
y0=±s−(x2
0+25
4)+5rx2
0+25
16.
1