Question

1. (a) If a chord of the parabola y

2 = 4ax is a normal at one of its ends, show that its
mid-point

lies on the curve

2(xx 2a) = y

2

a

+

8a

3

y

2

.

Prove that the shortest length of such a chord is 6a√3

(b) Find the asymptotes of the hyperbola

x

2

2y

2 + 2x + y + 9 = 0.

Answer #1

An equation of a hyperbola is given.
x^2/ 9 - y^2//49=1
(a) Find the vertices, foci, and asymptotes of the hyperbola.
(Enter your asymptotes as a comma-separated list of equations.)
vertex
(x, y)
=
(smaller x-value)
vertex
(x, y)
=
(larger x-value)
focus
(x, y)
=
(smaller x-value)
focus
(x, y)
=
(larger x-value)
asymptotes
(b) Determine the length of the transverse axis.
(c) Sketch a graph of the hyperbola.

find the vertices,foci and asymptotes of the hyperbola and
sketch its graph y^2/25-x^2/9=1
show all work and step

1. Let T(x, y, z) = (x + z, y − 2x, −z + 2y) and S(x, y, z) =
(2y − z, x − z, y + 3x). Use matrices to find the composition S ◦
T.
2. Find an equation of the tangent plane to the graph of x 2 − y
2 − 3z 2 = 5 at (6, 2, 3).
3. Find the critical points of f(x, y) = (x 2 + y 2 )e −y...

For the following problem, do not include a graph.
(a) For the parabola y=−2x^2+9x+1,
find the vertex.
(b) Does the vertex correspond to the highest point or the
lowest point on the graph?

Find the shortest distance from the point P = (−1, 2, 3) to the
line of inter- section of the planes x + 2y − 3z = 4 and 2x − y +
2z = 5.

Choose the correct answer:
1) The curve r = 2 + cos(2θ) is symmetric about Select one:
A_ all of these answers.
B_ the x-axis with vertical tangent at θ= π
C_ the origin with vertical tangent at θ = 2π
D_ the y-axis with vertical tangent at θ = 0.
2) The equation 3x^2 - 6x - 2y + 1= 0 is Select one:
A- a parabola open up with vertex (-1 , 1) and p = 1/6
B-...

1. Check that a curve x=cosht, y=sinht becomes a hyperbola. Draw
its graph, and calculate its curvature.
2. Let a be a positive constant number. Draw the graph of a
catenary y=acosh(x/a). Calculate the arc length s from the point
(0,a) to the point (x,acosh(x/a)), and find the expression of the
curve in terms of the parameter s.

3. Find the equation of the tangent line to the curve 2x^3 + y^2
= xy at the point (−1, 1).
4. Use implicit differentiation to find y' for sin(xy^2 ) − x^3
= 4x + 2y.
5. Use logarithmic differentiation to find y' for y = e^4x
cos(2x) / (x−1)^4 .
6. Show that d/dx (tan (x)) = sec^2 (x) using only your
knowledge of the derivatives of sine/cosine with derivative
rules.
7. Use implicit differentiation to show that...

2.Use separation of variables to solve (3x^2(1-y^2))/2y with
initial condition y(1)=2.
3.State the solution of the homogeneous ODE with roots of its
characteristic equation of r= 1,1,1,+-7i,3+-5i.
4.Consider the system of linear equations:
2x+6y+z=7
x+2y-z=-1
5x+7y-4z=9
solve this system using: a) Carmer's rule, b)Gauss-Jordan
elimination, c) an inverse matrix.

1-) Obtain expressions for the line and plane that are
tangential and normal respectively to the curve 3x^2 * y + y^2* z=
−2 and 2x*z−x^2*y=3 at the point (1,-1,1).

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