+614 use the information provided to write the general conic form equation of each circle.
1. three points on the circle, ( 7,-4),(-3,-4), and (6,-7)
please help, I do not understand how to dod this problem.
+500 Hey Jenny! See you got some conics to deal with!
Let's start with the general equation of a circle, the one that most people are taught:
(x - a)2 + (y-b)2 = r2, where (a,b) is the center of the circle, and r is the radius of the given circle.
Because the question gives us 3 points, we know they all must satisfy this equation.
Let's pick the pair of points (7,-4) and (-3,-4) to plug in(there's a good reason for picking this pair. Do you see why?)
We can first look at (7,-4). This gives us:
(7-a)2 + (-4-b)2 = r2
Let's then look at (-3,-4). It'll start to become clear why I picked these two points. We get:
(-3-a)2 + (-4-b)2 = r2
We realize that we can then set these two equations equal to each other, to get:
(7-a)2 + (-4-b)2 = (-3-a)2 + (-4-b)2
Now here's where our selection of points comes in handy. Realize we can just subtract
(-4-b)2 on both sides, which gives us only a, allowing us to solve for a without getting a nasty(ier) expression than if we chose another pair.
We get:
(7-a)2 = (-3-a)2
Expanding, we get:
49-14a + a2 = 9+6a + a^2
Combining like terms and subtracting appropriately, we get:
40 = 20a
a = 2
To get b, let's create another statement with two different points, say (7, -4) and (6,-7)
We get:
(7-2)2 + (-4 -b)2 = (6-2)2 + (-7-b)2
This gives us:
25 + (-4 -b)2 = 16 + (-7-b)2
9 + 16 + 8b + b2 = 16 + 49 + 14b + b2
Combining like terms and subtracting where is necessary, we obtain the equation:
6b = -40
b = -20/3
Our circle's equation is then:
(x - 2)2 + (y+20/3)2 = r2
I'll leave finding the radius to you(it should be fairly simple, just plug in one of the points to find it, and then take the square root of the result.
