Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Under what conditions on $a, b, c,$ and $d$ is $a x^{2}+a y^{2}+b x+c y+d=0$(a) a circle, (b) a point, (c) a line. (d) a contradiction?

(a) $b^{2}+c^{2}-4 a d>0$(b) $b^{2}+c^{2}-4 a d=0$(c) $a=0, b c \neq 0$(d) $b^{2}+c^{2}-4 a d<0$

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

Campbell University

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

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).

03:18

02:56

If $\bar{a}=b, c \in \math…

01:19

What relationship between …

03:38

If $|a|=|b| \neq 0$ and $\…

01:38

If one of the circles $x^{…

04:02

If $|a| \neq|b|$, then the…

05:38

If $\bar{a} \neq b$, then …

01:24

The line $\mathrm{x}-\math…

01:35

Determine whether the give…

01:52

for this problem. We have been given a generic equation. A, B, C and D are numbers. Those were my coefficients or my constant term and X and y are my two variables now for this problem. What we want to know is, how can we classify this? How do I know what shape is shown here? In particular, there's four shapes were going toe Look for we're gonna look to see what conditions would give me a circle. What conditions would give me a point, a line and a contradiction. That contradiction means that there is no graph in the real numbers that represents this particular equation. So those are my four, um, possibilities. Let's start with the third one first the line because that's the simplest one. Think back to what a line looks like. Um, a standard form for a line of slope intercept form. Why equals M X plus B. In this case, there is no X squared. There is no y squared. So in orderto have a line, I need a thio equal zero that would get rid of both my X squared and my wife square terms. One other thing we need for a line. We need to have either a. I mean sorry. Either B or C not equal to zero. I have to have at least an X or a Y term. Otherwise, I just have Ah, contradiction. I have a number equaling zero and have a line. I have to have either an X or a y or both. So either B or C or both. Um and I'm gonna have to rewrite that within or because I don't want that to be an ant. So either be or C does not equal sirrah. They could both have values. One could be a 01 could have a non zero value, but one of them has to exist, So that gives me a line. What about the other ones? Well, for those cases, let's take a look very briefly at the standard form for a line are for a circle standard form. First circle is X minus, H squared. Plus why minus k squared equals R squared. OK, if we can put this equation which we have a general form into standard form, this r squared will tell us what we have. A circle has an r squared this positive. Some positive number. You could take that square root. You can see what your radius is for your circle. If it shrinks down until there is no more radius, that means that our square is gonna equal zero. And that's a point. The Onley spot on my circle is that very Centrepoint? Um there's no radius to go out anywhere from there. We've already done our line. And a contradiction is if r squared is less than zero, every real number when you squared is positive. So if I have r squared is a negative, that doesn't correspond to any circle on the rial, Uh, under the real numbers. So that would be my contradiction. So we're going to need to take our general form and put it into standard form. Okay, so let's see what our steps are. If you look at the standard form, there is no coefficient in front of extra Why? So I'm gonna have Thio first of all, deal with a I don't wanna have a in front of X squared and y squared, so I have to get rid of that first, and then we have to complete the square. Okay, so those are two steps. So a first let's divide every term by a That's going to give me X squared plus y squared plus B over a X plus C over a Y plus D over a equals zero. Now we're going to complete the square. I'm going to collect my ex terms and my white terms because we have to complete the square twice, once for each variable that d over a. That's my constant. I'm going to subtract from both sides. Now let's complete the square. Let's start with X. I look at the X term. I take half of the coefficient, which is be over to a and we square it b squared over for a squared, and I have to add that to both sides. That gives me X Plus B over to a square. Now for the why. Again, I look at the Y term, take half of the coefficient, see over to a and we square it and that C squared over for a squared. We added to both sides that keeps our equation balanced. And so when I complete the square, that's why, plus c over to a squared. Now let's add up all the constant numbers On the right hand side, remember, A, B, C and D are all representing Constance. Well, I need to come and denominator Looks like that'll be for a squared. So I multiplied by four a Multiply the top by four a putting that together. And I like the positives first. I'm just gonna change the order here. B squared plus c squared minus four A. D over four. A squared. Okay, Now, we have already assumed we've already taken care of the case. When a is zero, that's a line. So in order, have a circle A is positive, Which means I don't have to worry about dividing by zero. That's gonna be some number in order to know and is always gonna be positive. A squared is always gonna be a positive Numbers of for a squared were great there. So, in order to know whether R squared is positive, negative or zero, I'm gonna look at the top piece here. Okay, So if b squared plus c squared minus four a. D, that's my r squared value. If that is positive that I have a circle, If this value equals zero, that's my second case, and I have a point. And if I have B squared plus C squared minus four a. D. If that is negative, that is my contradiction. So those are my four cases. If I haven't a value, is going to be one of those three cases, and that's how we determine what the graph looks like. And if a zero as long as I have either B or C not equal to zero, that will give me a line.

View More Answers From This Book

Find Another Textbook

01:04

In each of the following, solve the given quadratic equation exactly using t…

01:02

Determine the zeros and then, using the sign of the function, draw its sketc…

02:40

(a) Draw the graph of the parabola. (b) From your graph, estimate the $x$ -i…

02:43

Determine the equation of the parabola passing through the points (0,10)…

01:05

Determine the vertical asymptote(s) if one exists.$$f(x)=\frac{3 x^{2}}{…

You are given a pair of equations, one representing a supply curve and the o…

01:28

Determine the center and radius of the given circle and sketch its graph.

Determine if the function defined by the given equation is odd, even or neit…

00:55

Determine the horizontal asymptotes, if they exists.$$f(x)=\frac{2}{x-5}…

sketch the graph of the given ellipse, labeling all intercepts.$$\begin{…