SOLUTION: For each of the ff. quadratic equations, (a) identify the conic section represented (b) write the given equation in the standard form (c) give the properties (d) find the x and y-i

Question 1690: For each of the ff. quadratic equations, (a) identify the conic section represented (b) write the given equation in the standard form (c) give the properties (d) find the x and y-intercepts and (e) sketch the graph.
1. 2x^2-2y^2+20y-75=0
2. 9x^2+4y^2-36x+8y+31=0
I need the answers now. Thanks!
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
1. 2x^2-2y^2+20y-75=0
Sol: To complete the square : 2x^2-2(y^2-10y + (10/2)^2)=75-50 ,
So, 2x^2 - 2(y-5)^2 =25
Or x^2 - (y-5)^2 = 25/2
Or x^2/(25/2) - (y-5)^2/(25/2) = 1
a) This is a hyperbola center at (0,5)
b) Standard form : x^2/(25/2) - (y-5)^2/(25/2) = 1
c) a hyperbola centered at (0,5), its axis is y=5.
(I am notsure what you mean about properties.)
d) when y =0 , x^2 = 25/2 + 25 = 75/2, so x= sqrt(75/2) or -sqrt(75/2)
(two x-intercepts).
when x = 0, -(y-5)^2 = 25/2 has no solutions ,so no y-intercepts.
e) Graph of the hyperbola is left for you.(It's easy.)

2. 9x^2+4y^2-36x+8y+31=0
Sol: By complete square : 9(x^2-4x+4) +4(y^2+2y +1) =-31+36+4 ,
Or 9(x-2)^2 + 4(y+1)^2 = 9,
Or (x-2)^2 + (y+1)^2/(9/4) = 1,
a) This is an ellipse centered at (2,-1)
b) Standard form : (x-2)^2 + (y+1)^2/(9/4) = 1,
c) an ellipse centered at (2,-1), its long axis a=3/2 is on y=-1,
short axis b is on x=2.
(I am notsure what you mean about properties.)
d)when y =0 , (x-2)^2 = 1-4/9 = 5/9, so x = 2 + sqrt(5)/3, or
x = 2-sqrt(5)/3. (two x-intercepts)
when x =0, (y+1)^2 = 9/4(1-4). No solutions, so no y-intercepts.
e) Graph of the ellipse is left for you.
Kenny
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