Solve each problem. Find the equation of the line passing through...

Question

Solve each problem. Find the equation of the line passing through the points of intersection of the graphs of x^2 + y^2 = 20 and x^2 - y = 0.

Accepted Answer

Hey, everyone in this problem, we're asked to find the equation of a line that passes through the intersection of the following equation. We're given three X squared minus 32 Y is equal to zero and X squared plus Y squared is equal to 100. But you'll notice is that first equation we were given A is the equation of a parabola. And the second is the equation of a circle. So we're given four answer choices. Option A Y is equal to six. Option B eight X plus six, Y is equal to zero. Option C negative eight X plus six Y is equal to zero and option D X is equal to six. Now, we have these two equations and we want to figure out the equation of a line that passes through the intersection. So the first thing we need to do is find the intersection of these two equations. Now, we're gonna start with the equation X squared plus Y squared is equal to 100. And then we're gonna write out our other equation as well. Three X squared minus 32 Y is equal to zero. And we're gonna start with the substitution method. OK. We have two equations with two unknowns could solve this in any way you want. But we're gonna do the substitution method. And we're gonna start by rearranging our equation on the left hand side for the circle X squared plus Y squared equals 100 or X squared. OK. If we isolate for X squared, then our other equation has an X squared term and we can just substitute that rated. So rewriting, we have the X squared is equal to minus Y squared, right? And we're gonna call this equation one. Now we're gonna go to our other equation three X squared minus 32 Y equals zero. And we're gonna substitute in that equation one that we just wrote down for X squared. So we write this as three multiplied by minus Y squared minus 32 Y is equal to zero. Now, I notice that we have an equation of just why. OK. The only unknown we have is why we can go ahead and solve for why? So let's expand and simplify, OK. Expanding these brackets, we have 300 minus three Y squared minus 32 Y is equal to zero. I want to move everything to the right hand side just so that we have a positive leading coefficient. So we have three Y squared plus Y minus 300 is equal to zero. And we want to solve for why we're gonna go ahead and use our quadratic formula. So we call for a quadratic formula where the A, the B and the C come from the A is gonna be the coefficient corresponding to our Y squared term. So that's three B is gonna be the coefficient corresponding to our Y term. So that's 32 and C is gonna be our constant term which is negative 300. So, and the quadratic formula gives us that Y is equal to negative B lots or minus the square root of B squared minus four AC all divided by two A substituting in are A B and C values Y is equal to negative 32 plus or minus this square root of 32 squared minus four, multiplied by three, multiplied by negative 300 all divided by two multiplied by three. If we simplify, we get negative 32 plus or minus the square root of all divided by six. OK? Now the square root of 4624 that works out nicely to be 68. So yet that Y is equal to negative W minus divided by six. OK? And this is gonna give us two solutions and let's write those two solutions. The first solution which we're gonna call Y one. OK. This is gonna happen when we subtract. So we get negative 32 minus 68 divided by six which gives us negative 100 divided by six which we can simplify by dividing numerator and denominator by two to negative 50 divided by three. And our second Y value is gonna come from adding so negative plus 68 divided by six. OK. This gives us a Y two, it's equal to six. So we have these two Y values where we have the intersection of these equations. And what we wanna do is find the corresponding X values. So let's go back up to our first equation. Equation one. And we're gonna substitute in, OK. The first thing we're gonna substitute in is Y one which we found to be negative 50 divided by three. If we substitute that in, we get that X squared is equal to minus negative 50 divided by three squared. This gives us X squared is equal to minus 2500 all divided by nine. And you'll notice that the right hand side is gonna be negative a 900 minus 2500 gives us a negative value. So we have X squared is equal to a negative. That's gonna give us no real solutions. And there are no numbers where we can multiply them by themselves to get a negative. So we have no real solution. And so we actually don't get any intersection point from that particular Y value. So let's switch over to our second Y value. We're gonna substitute Y two which was equal to six. Now, into equation one, again, we get that X squared is equal to minus six squared. A six squared is equal to 36 100 minus 36. Gives us that X squared is equal to 64. And we get that X is equal to plus or minus eight. And I remember when you take the square root, you're gonna get both the positive and negative root. So we have that X is equal to plus or minus eight. All right. So we have two intersection points and we have negative eight comma six and we have positive eight comma six. And these are the two intersection points we get from our equations. Now, let's try a little diagram and kind of see what's going on here. OK. So we have our plane and we have, let's draw the circle first, we have X squared plus Y squared equals 100. OK. So this is a circle centered at the origin with radius 10. So we can draw a circle. We can imagine that's centered at the origin and it has a radius of 10. And then we have our other equation which is three X squared minus Y is equal to zero. And that is going to look something like this. And let me just redo that, that should be touching the origin there. So it was a little off so that should touch the origin because 00 is a point and it's opening upwards. So we have our Parabola and we have these two intersection points which we just found the one on the left is negative eight comma six, the one on the right is eight comma six. And our job is to find the line that cuts through these intersection points. OK. Well, if we draw a line connecting these points, we can see that it's just gonna be a horizontal line. OK? The Y value for each of these points is just six. So we get a horizontal line. Remember that the equation of a horizontal line is given by Y is equal to whatever the Y value is. And in this case, the Y value is just six. Our equation is Y is equal to six. OK? Now, if you didn't see that right away, if you didn't draw that diagram and realize that we just have this horizontal line, then you could substitute this into your equation. Y equals M X plus B. OK? You have two points you can go ahead and solve and you will also get the equation of the line Y equals six. So going up to our answer choices, OK? We found the intersection point of these equations and then we found the equation of the line that passes through those points. And we found that the correct answer is Y equals six which corresponds with option A. Thanks everyone for watching. I hope this video helped. See you in the next one.

Solve each problem. Find the equation of the line passing through the points of intersection of the graphs of x^2 + y^2 = 20 and x^2 - y = 0.

Key Concepts

Circle Equation

Parabola Equation

Finding Intersection Points

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