+152 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
+129852 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
+129852 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
