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This question asks you to find some information about a circle given the equation:
`(x+1)^2 + (y-6)^2 = 36`
Thankfully, this equation is almost in the form we need to find all of the information the question asks for.
Finding the center and radius
Here is the standard equation of a circle (see link below):
`(x-a)^2+(y-b)^2 = r^2`
Where `(a,b)` is the center of the circle and `r^2` is the square of the radius.
We just need to change one sign around and get 36 into the form 6^2 in order to get the form in the above formula, and we should have our center and radius:
`(x-(-1))^2 + (y-6)^2 = 6^2`
There it is, our center will be (-1, 6) and our radius is 6.
Finding the x and y intercepts
To find these intercepts, you must recall how the intercepts are defined. At an x-intercept, y = 0. At a y-intercept, x = 0.
To find the x-intercepts, we set y = 0 in our equation and solve for the x-values that allow y to be 0:
`(x+1)^2 + (0 - 6)^2 = 36`
Now, let's simplify by squaring the terms in the parentheses and combining "like" terms:
`x^2+2x+1+36 = 36`
`x^2+2x+37 = 36`` `
Now, we subtract 36 from both sides to get a quadratic equation:
`x^2+2x+1 = 0`
We could use the quadratic equation to find the x-values, but it is much easier to factor as we do below:
`(x+1)(x+1) = 0`
To solve this equation, we simply allow one or the other expression inside the parentheses to be equal to 0. This particular case is easy because both expressions are the same!
Solving by subtracting 1 from both sides:
`x = -1`
This gives us our only x-intercept (keeping in mind from before that y = 0):
To find the y-intercepts, we'll do pretty much the same thing. The difference is that where we intersect the y-axis, we have an x-value of 0, so we'll need to solve for the y-values. Let's put x = 0 into our equation:
`(0+1)^2 + (y-6)^2 = 36`
Now, we'll simplify again:
`1 + y^2-12y+36 = 36`
Again, we'll subtract 36 from both sides:
`y^2-12y+1 = 0`
Now, we have a different quadratic equation, which we'll have some trouble solving through factoring, so we'll just use the quadratic equation:
`y = (-(-12) +- sqrt((-12)^2 - 4(1)(1)))/(2(1))`
Evaluating this gives us the following 2 values for y:
y = 0.084 and y = 11.92
This gives us our y-intercepts:
(0, 0.084) and (0, 11.92)
As a summary:
`C = (-1, 6)`
(0, 0.084), (0, 11.92)
To see if these values are reasonable, let's graph the equation:
Believe it or not, that's a circle. The scale is just a bit off. However, if you make estimates based on the values we determined above, you'll see that they all fit pretty well with this graph. Hope that helps!
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