Expert: Sherman D. - 4/20/2004

Question
Intermediate Algebra

1. Write the equations for a system that contains a circle and an ellipse that do NOT intersect (without using degenerate curves).

2. Write the equations for a system that contains a circle and a hyperbola that do NOT intersect (without using degenerate cases).

3. You must solve both (a) and (b) correctly
solve the following system (a) {x^2+y^2-6x=0
                                                 {4x-y=9
                                             (b) {x^2+y^2=4
                                                   {x^2+y^2-4x-4y+4=0
4. The ONLY point of intersection of a diagonal line and the parabola y=x^2 is (-2,4).
Determine the equation of the line (in slope-intercept form)

5. For the hyperbola x^2/4-y^2/4=1, the equations of the asymptotes are y=x and y= -x. Prove algebraically (with explanation) that the hyperbola never touches (intersects) the asymptotes.

Answer
Sorry i can't really help you.

Try finding your answers at www.quickmath.com.

However i will try to answer some of your questions.

3.)
a.)
x^2 + y^2 - 6x = 0
4x - y = 9

-y = -4x + 9
y = 4x - 9

x^2 + (4x - 9)^2 - 6x = 0
x^2 + ((4x - 9)(4x - 9)) - 6x = 0
x^2 + (16x^2 - 36x - 36x + 81) - 6x = 0
x^2 + 16x^2 - 72x + 81 - 6x = 0
17x^2 - 78x + 81 = 0

Using quadratic formula

x = (-b ± sqrt(b^2 - 4ac))/2a

x = (-(-78) ± sqrt((-78)^2 - 4(17)(81)))/(2(17))
x = (78 ± sqrt(6084 - 5508))/34
x = (78 ± sqrt(576))/34
x = (78 ± 24)/34
x = (102/34) or (54/34)
x = 3 or (27/17)

y = 4x - 9
y = 4(3) - 9
y = 12 - 9
y = 3
or
y = 4(27/17) - 9
y = (108/17) - 9
y = (-45/17) or you can call that -2 and (11/17)

Ans :
x = 3
y = 3
or
x = 1 and (10/17)
y = -2 and (11/17)

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(b)
x^2 + y^2 = 4
x^2 + y^2 - 4y + 4 = 0

substitute x^2 + y^2 with 4, into the second problem

4 - 4x - 4y + 4 = 0
-4x - 4y + 8 = 0
-4y = 4x - 8
y = -x + 2

x^2 + (-x + 2)^2 = 4
x^2 + ((-x + 2)(-x + 2)) = 4
x^2 + (x^2 - 2x - 2x + 4) = 4
x^2 + x^2 - 4x + 4 = 4
2x^2 - 4x + 4 = 4
2x^2 - 4x = 0
x^2 - 2x = 0
x(x - 2) = 0
x = 0 or 2

y = -x + 2
y = -(0) + 2
y = 0 + 2
y = 2
or
y = -(2) + 2
y = -2 + 2
y = 0

Answer
x = 0
y = 2
or
x = 2
y = 0

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((x^2)/4) - ((y^2)/4) = 1
x^2 - y^2 = 4
-y^2 = -x^2 + 4
y^2 = x^2 - 4
y = ±sqrt(x^2 - 4)

as far as prooving, i am not good at that, but you can go to www.quickmath.com and type it in.