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Find the intersection of the line and the circle givenbelow.4x+y=-1x^2+y^2=10...

Question

Find the intersection of the line and the circle givenbelow.4x+y=-1x^2+y^2=10

Find the intersection of the line and the circle given below. 4x+y=-1 x^2+y^2=10



Answers

Find the coordinates of the points of intersection of the line $y=1$ and the circle centered at $(3,0)$ with radius 2.

When comparing circles and lines or the equations of circles and lines. At the same time, we know that a line can intersect a circle at two points. If the line is a Secret line, it can intersect at one point. If it's a tangent line or it may not intersect at all. And so let's take a look at the equation of a line in the equation of a circle and let's put them together. I'm gonna start by changing my linear equation into slope intercept form, and we're gonna substitute that for why an hour circle equation? So we have X squared, plus our wise negative X plus five. All squared equals 25 and next I want to multiplier binomial. So I have X squared gives me positive X squared minus 10 X plus 25 that those three terms come from multiplying your binomial Zell's equals 25. Ah, solving. I'm going to subtract 25 from both sides and simplify my ex square terms. So I have two x squared minus 10 X equals zero so we can factor out the X term. So that's two X gives me X minus five equals zero. So we have two X equals zero and X minus five equals zero. So our final solutions are X equals zero four X equals five. So we need to substitute them into our equations to see what are y values are and what the possibilities are them. So let's start by substituting them in our linear equation. So are linear equation when x zero Why minus equals negative zero plus five. So when x zero Why is five and then we have one X is five. So we have a y equals negative five plus five, which means when X is five, why is going to be zero? So it is possible that our line it could intersect the circle of thes two points. Eso Let's take a look at what happens when we substitute these X values into our circle equations. So when x zero we get zero squared equals R plus, y squared equals 25. So that means why is equal to five or negative five. The only issue with that is that's the 2.5 and zero negative five. And if are linear equation actually intersected? The circle at those two points are linear equation would be vertical, and we know that why equals negative. X Plus five is a diagonal line. It's got a negative slope to So if when x zero wise five is a solution for both of these equations, then when x zero wife was negative? Five cannot be a solution. Also, if I substitute five into our when X is five indoor equation, we get five squared plus y squared equals 25. Subtracting 25 from both sides. We get y squared equals zero. So we know that when x is five y zero. So it is the same two points for both equations. So those are our solutions Are point of intersections of the linear equation X plus White was five and the circle X squared plus y squared equals 25.

Here is we can see no matter how far we zoom out, we will not have the other equation for the circle that we're looking for. We just have this line that tells us that the there is no solution to the system equations.

Hello. So we want to find the intersection of the circle X squared plus y squared minus x minus three. Y equals zero. With the line, Y is equal to x minus one. So what we can do is just go ahead and take y equals x minus one substituted in for Y. In the equation of a of the equation given for the circle. And we end up with well X squared and then we get a plus x minus one squared and then minus x minus uh three times now X minus one is equal to zero. So if we um simplify this we get an x squared then we get x minus one squared, gives us an X squared. So we get a plus that X squared minus two X plus one. And then we get a minus x minus three here distributes and we get a minus three X plus three equal to zero. Combining like terms gives us two X squared minus six X plus four equal to zero. We can then uh we'll divide everything here by two to give us just x squared minus three, X plus two is equal to zero. We can in fact this nicely as x minus one times x minus two. So we get an x minus one times x minus two equal to zero. And we see clearly here that while either x minus one is zero or x minus +20 meaning that X is either equal to one um or two, therefore are two points of intersection Are well when x is one, why is zero? So the 00.1 comma zero and the point um in the point to um comma one, right when X is to um Y is going to be one, right going back to the equation um Y is equal to x minus one. So the 10.10 and a 0.2 comma one are going to be our points of intersection, as we can see hear from the graph. So here is our circle, here is our line. Why if you go to x minus one and we see that intersect right here at the 10.1 common zero and right here at the 00.2 comma one. Yeah.

In discussion. We have to find an equation off a line which is passing through centers off two circles, and the equation off the circle is given us X squared plus y squared minus four X plus. Six y plus four equals zero and they question off. The second circle is given us X squared plus y squared plus six x plus. Four y plus nine equals zero. Now we have to find centers off Bordeaux circles. For that, we will write that given equations in the standard form often equation. So let's right the first equation as X squared minus four x plus y squared plus six y equals minus four. Now for completing the square off X squared minus four X, we will add four on board the sides of the equation and for completing the square of Ice Core plus six. Why we will add nine on both the sides off the equation. Now, By simplifying this, we will get X minus two. Hold square plus y plus three. Whole square is equally toe nine. Now, by comparing this equation with the standard form off in a question which is X minus h hold script plus y minus K. Hold squared is equal to our square were h Comma K is the center So here Centerville big wayto two comma minus three. Now, similarly, we will find center off the second circle. For that, we will relight the equation as extra plus six x plus y squared plus four way equals minus nine. Now for completing the square off extra plus six X, we will at nine on the both sides of the equation and for completing the square of ice scripless for why we relied four on both sides of the equation. Now, by simplifying this, we will get X Plus three whole square plus y plus two, whole square is equate +04 Now again, By comparing this with the standard form often in question, we will get sent direct minus three comma minus two. Now we have to find any creation off a line which is passing through no comma minus three, which is center of the first circle, and to the point minus trico ma minus two, which is center off the second circle. First, we will find slope off the line, which will be equal to minus two minus off minus three divided by minus three minus two. That is why co dinner off the second point, minus y coordinate of the first point, divided by X. Coordinate off the second point minus X Coordinate off the first point. Now, by simply find this, we will get slow pick well toe minus one, divided by five. Now, by using the points low form, we will find an equation off line. And that is why I, minus y one, is equal to M times X minus X one. Where am Mr Slope and extra income of Ivan is anyone? Point off the line. So let's use the 10.0.2 comma minus three. We will get vie Plus three is equal to minus one, divided by five multiplied with X minus toe. Now by simply find this, we will get why is equal to minus X. Divided by five minus 13. Divided by five


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