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Determine the center and radius of the given circle and sketch its graph.$$16 x^{2}+16 y^{2}-48 x+8 y-27=0$$

$$\mathrm{C}(3 / 2,-1 / 4) \mathrm{r}=2$$

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

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University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Determine the center and r…

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Find the center and the ra…

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Find the center and radius…

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for this problem, we are going to be examining the equation. 16 X squared plus 16 y squared minus 48 X plus eight y minus 27 equal zero. This is the equation of a circle. So in order to graph this, the two pieces of information we need to find are the center and the radius. Now, if a circle is given to us in standard form, this is easy because we can see just by inspecting it both the center and the radius as a review, standard form for a circle is X minus h squared. Plus why minus k squared equals r squared. So this shows us the radius. It's the square root of the constant and it shows us the center, which is going to be the point h k the numbers I'm subtracting from x and Y respectively. So if we're given this format, graphing a circle is very straightforward. Unfortunately, we have been given our circle in general form, which means has been set equal to zero. So our goal here is going to be to take our general form equation and change it to standard form. Okay, step one in our standard form X and Y are by themselves. There's no coefficient, it's just X minus. Why minus. So I want to get rid of those 16. I'm going to divide every term in my equation by 16. That gives me X squared plus y squared, which is what I want minus three X while 8/16. That's a half. And this does not cancel it all. I get 27/16 equals zero. Now, in order to get my general form now to look like standard form, I have to complete the square once for the exes. So I'm gonna combine my ex terms and once for the why. So I'm going to combine my Y terms and that negative 27 16th. I'm gonna add to both sides to move it over to the right hand side. Now let's complete the square excess first. Um, I'll take the half of the coefficient of my ex term, which in this case will be negative three halfs and I square it so that becomes 9/4. I have to add it to both sides of my equation to stay balanced, so this gives me X minus three halves squared now from my why. Again, Let's look at the Y term. I take half of that coefficient. Half of a half is 1/4. I square that that becomes 1/16. And I can add that to both sides of my equation. Yeah, I end up with Y plus 1/4 square that completes the square for the wise. Now I just need to finish putting together my Constance. I do need a common denominator, so I'll multiply this middle one top and bottom by four. And when I add those that gives me 64/16, in other words, it gives me four. So now I'm in standard form. My radius squared is four square root of that gives me a radius of two, and I look at what I'm subtracting from X and y to find my center point. I'm subtracting three halves from X, and it's why, plus 1/4 which means I'm subtracting a negative 1/4. So there's my center. Three halves. Negative. 1/4. If I want to plot that, it's going to be about their on my graph. Three halves negative 1/4 and my radius is too, so I can go two units in each of the cardinal directions. That gives me four dots that I can do approximately a smooth curve to sketch my circle. This has a center at three halves, negative 1/4 and it has a radius of two.

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