SM2 14.1-8: Given Two Endpoints of a Diameter, Write the Cirlces Equation

Añadido: 2026-05-10

[00:00:00]Hello and welcome. So given that you have 16, negative 12 and negative 4, 6 being the endpoints of a certain diameter in a circle,  write the equation. Yes I blacked out the earlier ones they were a weak example we don't do weak  examples here, so let's get going. First off we've got to know the general equation of a circle: x  minus h squared plus y minus k squared equals your radius squared. So looking at this gem we've  got to know three things: got to know your h, your k, and your r. Well if you know the center  of your circle that's h comma k. So we can find the center of our circle, we know two out of the  three things that we need to know and we can just find the third, uh the radius, not not too hard  for this one. Alrighty so let's dive right into it. Let's talk about a diameter and a couple of  relationships. First off if you have any circle, oh wow that's not a circle clearly, okay any  circle and inside that circle you have a diameter.

[00:00:57]That means that for the center of a circle there  is a line that goes from one side of the circle to the other passing through that center. And there's  a cool relationship that if you have the two endpoints of a diameter then that means that the  average or essentially the middle of your two x's, right, is going to be the center. Likewise you  can do the same thing for the y's, right? If you take the middle of these two y's you'll find the  center. And that's exactly what we're going to do to find our h and k here. Where we know that if  we take the middle of our two points, so we have 16 and negative 4, right? If we add those together  and take half of it that should be where our h is.

[00:01:38]Okay, likewise if we take our two y values, right?  Negative 12 and 6. So we have negative 12 and 6, add them together and half that, find the middle,  then that should be the center of our circle, right? So on this one 16 plus a negative 4 that's  the same thing as 16 minus 4, that's 12, and half of 12, 6. That means that the h value, the direct  middle between 16 and negative 4 is positive 6.

[00:02:10]Okay, likewise for k, um negative 12 plus 6 that's  negative 6. Half of negative 6, negative 3. That means the middle between negative 12 and 6 is  negative 3. And it is right because if we go nine, add nine and subtract nine, we'll end up  at negative 3 both ways. Alrighty so now we know our h and k and that's most of the hard  work for this, right? That means that we know our equation is mystically already at x minus  our h which is 6 squared plus y minus our k, negative 3, squared equals our radius squared. And  we can simplify that a little bit more guys. What is subtracting a negative? What's that the  same as? Addition. Okay, so then you have x minus 6 squared plus y plus 3 squared equals  r squared. Now the only thing we want to find is our r squared. That's the only thing we're  missing, our radius. How can we find it? Well all we need to do is we have three variables,  right? We have x, we have y, we have r. So that means if we can find out what two out of our  three variables are we can find the last one just by solving. Well we know, we know what x and y are.  In fact we know it twice. We know that we have a solution that when x is 16, y is negative 12. We  have another one that when x is negative 4, y is 6. Bam, right? Plug it in. So I'm going to use the  negative 4, 6. Could you use the other one?

[00:03:37]Heck yeah, choose whichever point you want but  I'm going to just use negative 4,6 because why the heck not, right? That's your x comma y.  So that means x is negative 4 and your y, 6, equals r squared. And then from there all  you're doing is that you notice that r is the only variable left. Solve for it, right?  Simplify your numbers on the left. So we have negative 4 minus 6 that's negative 10. 6 plus 3  that's 9. And then we have equals r squared. So negative 10 squared is 100. 9 squared 81 equals  r squared which means r squared equals 181.

[00:04:19]Now do we have to simplify this? No because  think about it this way. The only thing you need in your equation is you need to know r  squared. You're not asking what the radius is. We really need to know what the radius  squared is so we don't have to do anything further to this. That means your final answer  is going to be x minus 6 squared plus our y plus 3 squared equals 181 because all we're doing  is we're just plugging that r squared into our original equation. And that's it. Just that  quick substitution. Alrighty thanks for watching.