Question

1. Write the equation in standard form. Identify the important features of the graph:

x^2+y^2-9x+10y+15=0

2. Write the equation in standard form. Identify the important features of the graph:
y^2+4y-8x+4=0

3. Write the equation of the conic section with the given properties:
A hyperbola with vertices, (0, -6) and (0,6), asymptotes, y=3/4 x and y = -3/4 x

4. Write the equation of the conic section with the given properties:
An ellipse with vertices, (0,-5) and (0,5), and a minor axis of length 8.

Answer 1

1. (x-4.5)^2 +(y +5)^2 = 30.25
2. x = (1/8)y^2 +(1/2)y +(1/2)
3. y^2/36 -x^2/64 = 1
4. x^2/16 +y^2/25 = 1

Explanation:

1. Complete the square for both x and y by adding a constant equal to the square of half the linear term coefficient. Subtract 15, and rearrange to standard form.

(x^2 -9x +4.5^2) +(y^2 +10y +5^2) = 4.5^2 +5^2 -15

(x -4.5)^2 +(y +5)^2 = 30.25 . . . . . write in standard form

Important features: center = (4.5, -5); radius = 5.5.

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2. To put this in the form x=f(y), we need to add 8x, then divide by 8.

x = (1/8)y^2 +(1/2)y +(1/2)

Important features: vertex = (0, -2); focus = (2, -2); horizontal compression factor = 1/8.

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3. We want y^2/a^2 -x^2/b^2 = 1 with a=36 and b=(36/(3/4)^2) = 64:

y^2/36 -x^2/64 = 1

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4. In the form below, "a" is the semi-axis in the x-direction. Here, that is 8/2 = 4. "b" is the semi-axis in the y-direction, which is 5 in this case. We want x^2/a^2 +y^2/b^2 = 1 with a=4 and b=5.

x^2/16 +b^2/25 = 1

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The first attachment shows the circle and parabola; the second shows the hyperbola and ellipse.