*Please*-Need Help:Algebra 2 Conics Question(s)

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edd fedds

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~Please help me with these confusing problems I gott stuck on. I greatly appreciate it.
1.)Find radius of a circle with the equation x^2+y^2-6x-12y+36=0?
a.9
b.6
c.3
d.1
*I first changed the equation to general form, using the formula (x-h)^2+(y-k)=r^2. My result was that r=9.17, so I choose a. Am I right?
2.)Find the center of an ellipse with the equation 9x^2+16y^2-18x+64y=71.
a.(-1,2)
b.(1,-2)
c.(-2,1)
d.(2,-1)
*I gott stuck on the third step in putting the equation in standard form first, which was 9(x^2+2x+?)-16(y^2+4y+?)=71+9(?)+16(?). Really need help.
3.)Find the major and minor axis of an ellipse with the equation 16x^2+25y^2+32x-150y=159.
a.5 & 4 units.
b.25 & 16 units.
c.10 & 8 units.
d.50 & 32 units.
*I have no idea on how to start solving this problem, given the form of the equation.
4.)Find the slopes of the asymptotes of a hyperbola with the equation y^2=36+4x^2.
a.2,-2.
b.1/2,-1/2.
c.4,-4.
d.1/4,-1/4.
*Do not know how to change this equation to standard form in order to solve the equation.
5.)What is the graph of the equation x^2+2x+7-y=0
a.parabola
b.circle
c.ellipse
d.hyperbola
*I changed the equation to x^2+0xy+2x+7-y=0. Then I solved for A,B,& C OR A=1, B=0, & C=2. Then I used the equation B^2-4AC and gott d=-8. Which graph would this be?
6.)Identify the solutions to the system of equations.
x^2+y^2=25
16x^2+25y^2=400
a.no solutions
b.(5,0)
c.(-5,0)
d.(5,0) and (-5,0)
*How would I go about solving this?
:mrgreen:
 
~Please help me with these confusing problems I gott stuck on. I greatly appreciate it.
1.)Find radius of a circle with the equation x^2+y^2-6x-12y+36=0?
a.9
b.6
c.3
d.1
Click to expand...

I will show you how to do this one and maybe it'll help with the others. Okey-doke?. :D

We complete the square by grouping the x's and y's by taking the square of half the coefficient of x and adding to both sides.

\(\displaystyle (x^{2}-6x+9)+(y^{2}-12y+36)=-36+9+36\)

Factor:

\(\displaystyle (x-3)^{2}+(y-6)^{2}=9\)

The circle has center (3,6) and radius 3.

*I first changed the equation to general form, using the formula (x-h)^2+(y-k)=r^2. My result was that r=9.17, so I choose a. Am I right? Sorry to say, no. You have the idea, but must have made an algebraic or arithmetic mistake. The radius is a nice integer, not 9.17

2.)Find the center of an ellipse with the equation 9x^2+16y^2-18x+64y=71.
a.(-1,2)
b.(1,-2)
c.(-2,1)
d.(2,-1)
Click to expand...

Do the same thing....complete the square.

\(\displaystyle 9(x^{2}-2x)+16(y^{2}+4y)=71\)

Complete the square:

\(\displaystyle 9(x^{2}-2x+1)+16(y^{2}+4y+4)=71+9+64\)

\(\displaystyle 9(x-1)^{2}+16(y+2)^{2}=144\)

Now, divide by 144 to get 1 on the right side to get the ellipse form.

\(\displaystyle \frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{9}=1\)

The ellipse is centered at (1,-2), has semi-major axis length 4 and semi-minor axis length 3.

\(\displaystyle \frac{(x-1)^{2}}{4^{2}}+\frac{(y+2)^{2}}{3^{2}}=1\)

Now, there's two of them. Hopefully this will allow you to finish the others.

4.)Find the slopes of the asymptotes of a hyperbola with the equation y^2=36+4x^2.
a.2,-2.
b.1/2,-1/2.
c.4,-4.
d.1/4,-1/4.
*Do not know how to change this equation to standard form in order to solve the equation.
Subtract 4x^2 from both sides and divide by 36, simplify.
Click to expand...

Set your equation up in the form of a hyperbola.

\(\displaystyle \frac{y^{2}}{36}-\frac{x^{2}}{9}=1\)

The equations of the tangents are simply \(\displaystyle y=\pm\frac{a}{b}x\)
 
3.)Find the major and minor axis of an ellipse with the equation 16x^2+25y^2+32x-150y=159.
a.5 & 4 units.
b.25 & 16 units.
c.10 & 8 units.
d.50 & 32 units.
*I have no idea on how to start solving this problem, given the form of the equation.

16x^2 + 25y^2 + 32x - 150y = 159
16x^2 + 32x + 25y^2 - 150y = 159 (Rearranging the terms)
16(x^2 + 2x) + 25(y^2 - 6y) = 159 (Factoring out the coefficients of x^2 and y^2)
16(x^2 + 2x + 1) + 25(y^2 - 6y + 9) = 159 + 16 + 225 (Completing the squares)
16(x + 1)^2 + 25(y - 3)^2 = 400 (Factoring the perfect square trinomials)
(x + 1)^2/25 + (y - 3)^2/16 = 1 (Dividing both sides by 400)
a^2 = 25 and b^2 = 16
a = 5 and b = 4
The major axis is 10 (2 • 5).
The minor axis is 8 (2 • 4).

4.)Find the slopes of the asymptotes of a hyperbola with the equation y^2=36+4x^2.
a.2,-2.
b.1/2,-1/2.
c.4,-4.
d.1/4,-1/4.
*Do not know how to change this equation to standard form in order to solve the equation.

y^2 = 36 + 4x^2
y^2 - 4x^2 = 36 (Isolating the constant on the right side)
y^2/36 - x^2/9 = 1 (Dividing by 36)
a^2 = 36 and b^2 = 9
a = 6 and b = 3
Notice the y^2-term is positive and the x^2-term is negative.
The hyperbola opens horizontally.
When you draw the asymptotes, you will draw a rectangle 6 units long and 12 units high.
The slope of the asymptotes is ±2 (12 ÷ 6).


5.)What is the graph of the equation x^2 + 2x + 7 - y = 0
a.parabola
b.circle
c.ellipse
d.hyperbola
*I changed the equation to x^2+0xy+2x+7-y=0. Then I solved for A,B,& C OR A=1, B=0, & C=2. Then I used the equation B^2-4AC and gott [sic] d=-8. Which graph would this be?

Actually, A = 1, B = 0, and C = 0. B^2 - 4AC = 0^2 - 4(1)(0) = 0

The curve is a parabola.

Also, note that the coefficient of y^2 = 0, so there is only one variable that is squared. PARABOLA!

6.)Identify the solutions to the system of equations.
x^2+y^2=25
16x^2+25y^2=400
a.no solutions
b.(5,0)
c.(-5,0)
d.(5,0) and (-5,0)
*How would I go about solving this?

x^2 + y^2 = 25
16x^2 + 25y^2 = 400

Multiply the first equation by -16 and add it to the second.

-16x^2 - 16y^2 = -400
16x^2 + 25y^2 = 400

9y^2 = 0
y^2 = 0
y = 0

x^2 + 0^2 = 25
x^2 = 25
x = ±5

(±5, 0) are the solutions, i.e. intersection points.
 
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