MATH 200 Lecture Notes - Lecture 12: Parametric Equation, Jyj, Paraboloid
Math 200 Problem Set II
1) Find the equation of the sphere which has the two planes x+y+z= 3, x +y+z= 9
as tangent planes if the centre of the sphere is on the planes 2x−y= 0,3x−z= 0.
2) Find the equation of the plane that passes through the point (−2,0,−1) and through
the line of intersection of 2x+ 3y−z= 0, x −4y+ 2z=−5.
3) Find the equations of the line through (2,−1,−1) and parallel to each of the two planes
x+y= 0 and x−y+ 2z= 0. Express the equations of the line in vector and scalar
parametric forms and in symmetric form.
4) Sketch and describe the following surfaces.
a) 4x2+y2= 16 b) x+y+ 2z= 4 c) z2=y2+ 4
d) x
4=y2
4+z2
9e) y2
9+z2
4= 1 + x2
16 f) z=x2
g) z=y2
4−x2
9h) y2=x2+z2i) x2
9+y2
12 +z2
9= 1
j) x2+y2+z2+ 4x−by + 9z−b= 0 where bis a constant.
5) Sketch the graphs of
a) f(x, y) = sin x0≤x≤2π, 0≤y≤1
b) f(x, y) = px2+y2
c) f(x, y) = |x|+|y|
6) Sketch some of the level curves of
a) f(x, y) = x2+ 2y2
b) f(x, y) = xy
c) f(x, y) = y
x2+y2
d) f(x, y) = xe−y
7) Describe the level surfaces of
a) f(x, y, z) = x2+y2+z2
b) f(x, y, z) = x+ 2y+ 3z
c) f(x, y, z) = x2+y2
8) Find the velocity, speed and acceleration at time tof the particle whose position is
~r(t). Describe the path of the particle.
a) ~r(t) = acos tˆı+asin tˆ+ct ˆ
k
b) ~r(t) = acos tsin tˆı+asin2tˆ+acos tˆ
k
MATH 200 PROBLEM SET II SOLUTIONS
1) Find the equation of the sphere which has the two planes x+y+z= 3, x +y+z= 9 as tangent
planes if the centre of the sphere is on the planes 2x−y= 0,3x−z= 0.
Solution. The planes x+y+z= 3 and x+y+z= 9 are parallel. So the centre lies on x+y+z= 6
(the plane midway between x+y+z= 3 and x+y+z= 9) as well as on y= 2xand z= 3x. Solving,
y= 2x, z = 3x, x +y+z= 6 ⇒x+ 2x+ 3x= 6 ⇒x= 1, y = 2, z = 3
So the centre is at (1,2,3). The normal to x+y+z= 3 is (1,1,1). The points (1,1,1) on x+y+z= 3
and (3,3,3) on x+y+z= 9 differ by a vector, (2,2,2), which is a multiple of this normal. So the
distance between the planes is k(2,2,2)k= 2√3 and the radius of the sphere is √3. The sphere is
(x−1)2+ (y−2)2+ (z−3)2= 3
2) Find the equation of the plane that passes through the point (−2,0,−1) and through the line of
intersection of 2x+ 3y−z= 0, x −4y+ 2z=−5.
Solution. First we’ll find two points on the line of intersection of 2x+ 3y−z= 0, x −4y+ 2z=−5.
This will give us three points on the plane.
2x+ 3y−z= 0
x−4y+ 2z=−5⇐⇒ 2x+ 3y=z
x−4y=−2z−5⇐⇒ 2x+ 3y=z
11y= 5(z+ 2)
In the last step, we subtracted twice the second equation from the first. So if z=−2, then y= 0 and
x=−1. And if z=−15
2, then y=−5
2and x= 0. So we conclude that the three points (−2,0,−1),
(−1,0,−2) and (0,−5
2,−15
2) must all lie on the plane. So the two vectors (−2,0,−1) −(−1,0,−2) =
(−1,0,1) and (0,−5
2,−15
2)−(−1,0,−2) = (1,−5
2,−11
2) must be parallel to the plane. So the normal to
the plane is (−1,0,1) ×(1,−5
2,−11
2) = 5
2,−9
2,5
2or, equivalently ˆn= (5,−9,5). The equation of the
plane is
5(x+ 2) −9y+ 5(z+ 1) = 0 or 5x−9y+ 5z=−15
3) Find the equations of the line through (2,−1,−1) and parallel to each of the two planes x+y= 0 and
x−y+ 2z= 0. Express the equations of the line in vector and scalar parametric forms and in symmetric
form.
Solution. One vector normal to x+y= 0 is (1,1,0). One vector normal to x−y+ 2z= 0 is (1,−1,2).
The vector (1,−1,−1) is perpendicular to both of those normals and hence is parallel to both planes.
So (1,−1,−1) is also parallel to the line. The vector parametric equation of the line is
~x = (2,−1,−1) + t(1,−1,−1)
The scalar parametric equations of the line are
x= 2 + t, y =−1−t, z =−1−t
The symmetric equations are
t=x−2 = −y−1 = −z−1
1
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MATH 200 Full Course Notes
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Document Summary
Express the equations of the line in vector and scalar parametric forms and in symmetric form: sketch and describe the following surfaces. 2: z2 = y2 + 4, z = x2. 4 = y: x, z = y, x2 + y2 + z2 + 4x by + 9z b = 0 where b is a constant, x + y + 2z = 4, y. 4 = 1 + x: y2 = x2 + z2. 2: sketch the graphs of, f (x, y) = sin x, f (x, y) = px2 + y2, f (x, y) = |x| + |y| 0 x 2 , 0 y 1: sketch some of the level curves of, f (x, y) = x2 + 2y2, f (x, y) = xy, f (x, y) = y. 2+y: f (x, y) = xe y x. The planes x + y + z = 3 and x + y + z = 9 are parallel.
