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avatar+144 

1)A parabola with equation y=ax^2+bx+ccontains the points (3,-3), (1,3), and (0,0) Find the value 100a+10b+c

2) In terms of pi, what is the area of the circle defined by the equation 2x^2+2y^2+10x-6y-18=0

I'm sorry for giving out a lot of questions, I am studying for AMC

 Dec 27, 2019
 #1
avatar+109563 
+2

1)

 

Since (0,0)  is on the graph, c   = 0

 

So we have that

 

a(3)^2  + b(3)  = -3     ⇒  9a  +  3b  = -3   ⇒    -3a - b  =  1    (1)

a(1)^2  + b(1)  =  3     ⇒   a  +  b    =  3           (2)

 

Add(1)  and (2)  and we get that

 

-2a  =  4

 

a = -2

 

b = 5

 

100 a +  10 b +  c   =  -200 + 50  + 0   =   -150

 

 

cool cool cool

 Dec 27, 2019
 #2
avatar+109563 
+1

2) In terms of pi, what is the area of the circle defined by the equation 2x^2+2y^2+10x-6y-18=0

 

Divide every term by 2  and rearrange as

 

x^2 + 5x  + y^2  - 3y  =  9      complete the square on  x  and y

 

x^2 + 5x + 25/4  +  y^2 - 3y + 9/4   =   9 + 25/4  +  9/4        factor and simplify

 

(x + 5/2)^2  +  ( y - 3/2)^2   =  36/4 + 25/4 + 9/4

 

(x + 5/2)^2  + ( y-3/2)^2  =  70 /4  = 35 / 2   =17.5

 

The area  =  pi * r^2   =    pi * 17.5    =  17.5 pi

 

 

cool cool cool

 Dec 27, 2019

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